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Kimia Analitik Lanjut

Lecturer:

A. Mutalib MSc, PhD



IntroductionThree categories of definitions of analytical chemistry:

1. Analytical chemistry produces information by application of available

analytical procedures in order to characterize matter by its chemicalcomposition. This refers to the actual production of analytical results and

requires instruments, procedures, and skilled personnel.

2. Analytical chemistry studies the process of gathering information by using

principles of several disciplines in order to characterize matter or systems.This covers the R&D of analytical procedures.

3. Analytical chemistry produces strategies for obtaining information by the

optimal use of available procedures in order to characterize matter of

systems. This can be considered as an organizational level comprising theinteraction between human and machines, including communication as we

as the optimal use of the tools available for producing information.



The quality of a sample is given by the similarity between the reconstructi

of the composition of an object and the object itself, as far as it is influenc

by the sampling strategy. It depends on the characteristics of the object and

the purpose of the reconstruction. For a mere description of the object, the quality can be expressed in the

sample quality.

For monitoring of the object, the sampling quality can be expressed as the

probability that a threshold crossing will be detected.

For object control the quality can be expressed as controllability ormeasurability.



The limit of detection is a quality parameter pertaining to analysis. It gives

the minimum concentration of a component that can be detected. It is

influenced by the absolute value of the blank, the standard deviation of themethod analysis, a safety factor. The lowest possible limit of detection is s

by the characteristic method.

The sensitivity gives the change of a signal upon a change of concentration

The sensitivity of a method is a practical quality measure, since it pertainsthe ease of detection. However, both the detection of a difference in

concentration between two samples and the limit of detection are ultimatel

governed by the precision of the analytical method and not by sensitivity. T

optimal value of the sensitivity should be such that the standard deviation

the method can be measured.

Quality Parameter



Quality Parameter

Selectivity and specificity can be expressed as the ratio of sensitivities of t

method of various components to be measured in the sample. The optimal

value of these characteristics is set by the number of measurements requireto estimate the composition of the sample.

Safety. Perhaps the cost of measures to be taken to satisfy safety regulation

can be used to measure safety as a quality parameter. The optimum is

determined by the balancing of adverse consequences and precaution cost.

Cost can be expressed in money or in other measures, such as manpower.

influenced by analysis time, standard labor, depreciation, and cost of

instruments, energy, and reagents. The optimal cost of an analytical metho

results from a comparison of various methods of analysis with respect toyield of the object and cost of analysis.



Quality Parameter

Information is defined as the difference in uncertainty before and after an

experiment. It is expressed as the binary logarithm of this uncertainty and

measured in bits.The information content of a qualitative analytical method is governed b

the selectivity and specificity of the method and the a priori knowledge

the occurrence of the sought component.

The information content of a quantitative analytical method is a functio

of the standard deviation of the method and the a priori knowledge of thrange compositions that can be expected.

The information content of a retrieval method is governed by the

selectivity of the method and the chance of occurrence of the sought item

Therefore information cannot be a quality criterion for a method as such

can be applied only in a specified situation. Correlation of items makes a priori information partly predictable and thus diminishes the informati

content of the experiment.



Methods Curve fitting is not a quality parameter, but it may enhance the quality of

measured data considerably for two reasons:

1. Interpolating the gap between data allows easier interpretation of many phenomena, although the information does not increase.

2. A most important use of curve fitting is in the unraveling of mixed

phenomena, for example, spectra.

Multivariate analysis may be a tool in establishing new quality criteria. Thfirst application may be in discovering patterns in the data produced by

analytical chemists. This enhances the value of these results. A measure fo

this effect is not known, but a “patterned” result seems to increase the quaof the analytical results.

Optimization is not a quality parameter. However, it may be applied in

optimizing analytical methods and thus improving quality.



Techniques to influence or may influence quality Sampling as a means to influence quality. Handling of samples can be of

great importance for some quality aspects of a sample. Homogenizing

influences the quality of the subsample, and so does sample reduction.Preservation and labeling influence both the identity and integrity of the

sample, an immeasurable quality aspect.

Selection of an analytical method may be based on one or more quality

criteria, and therefore these aspects should be weighed a priori. Techniquefor selection are, for example, the ruggedness test, the ranking test, pattern

recognition, optimization, and artificial intelligence.

Repeatability is the measure of variations between test results of successiv

test of the same sample carried out under the same conditions, i.e. the samtest method, the same operator, the same testing equipment, the same

laboratory, and short interval of time..



Techniques to influence or may influence quality Reproducibility is the term used to express the measure of variations betw

test results obtained with the same test method on identical test samples

under different conditions, i.e., different operators, different testing

equipment, different laboratories and/or different time.

Precision control and accuracy control. Control chart are means to trace

quality in time and indicate when intervention is required. Round-robin tes

may reveal shortcomings in precision and accuracy and, though complicat

psychological and organizational influences play a role, quality can impro

after test. Sequential analysis as well as analysis of variance may be used t

control precision and accuracy. A number of methods can be used to impro

precision by removing or diminishing irrelevant data (noise). Some

techniques treated are curve fitting and smoothing, Kalman filtering, and

deconvoluting filtering.

Planning

Organization.

li



Sampling

Sampling is often called the basis of analysis:

“The analytical result is never better than the sample it is based on”

Every part of the analytical procedure is important, because the final result

influenced by mistakes or added noise in every part procedure.

The purpose of sampling is to provide for a specific aim of the client part

the object that is representative of it and suitable for analysis. The quality of the sampling depends on many parameters, governed by

properties of the object and of the analytical procedure.

The properties of the object can be described in terms of physical

structure, size, and inhomogenicity in space and time.

As a rule the analytical chemist cannot or will not use the whole object in

his/her analysis machine but uses only a small part of the object: ~ 0.01 – 0.1



The sample, as a small fraction of the object to be investigated, must fulfill a

series of expectation before it may be called a sample and used as such. It m

• Represent faithfully the properties under investigation: composition, colo

crystal type, etc.

• Be of a size that can be handled by the sampler

• Be of a size that can be handled by the analyst, say, from 0.001 to 1.0 g

• Keep the properties the object had at the time of sampling, or change its

properties in the same way as the object

• Keep its identity throughout the whole procedure of transport and analys

To satisfy these demands, the analyst can use the results of much

theoretical and practical work from other disciplines.

Statistics and probability theory have provided the analyst with the theoretic

framework that predicts the uncertainties in estimating properties of populatiowhen only a part of the population is available for investigation.



The properties of analytical procedure can be described in terms of minim

and maximum sample size, physical structure of the sample, homogeneity

the sample in space, and stability in time, for example.

The requirements set by the object and by the analyst must be fulfilled, so sampler has to optimize the sampling parameters such as size, time or

distance spacing, number of subsamples, cost, mixing, and dividing.

The estimation of the value of the properties is the aim of the analytical

procedure. A feature of the object to be known is a list of the desired properties, for these determine the way of sampling.

The object can have many properties, for examples, composition, particle

size, color, and taste, but only selection of these properties is required to

describe the object quality.

Gl f t d i li



Glossary of terms used in sampling

Bulk sampling – sampling of a material that does not consist of discrete,

identifiable, constant units, but rather of arbitrary, irregular units.

Gross sample (also called bulk sample, lot sample) – One or more increme

of material taken from a larger quantity (lot) or material for assay or purposes.

Homogeneity – the degree to which a property substance is randomly

distributed throughout a material. Homogeneity depends on the size of the

units under consideration. Thus a mixture of two minerals may be

inhomogeneous at the molecular or atomic level but homogeneous at the particulate level.

Increment – an individual portion of material collected by a single operatio

of a sampling device, from parts of a lot separated in time or space.

Incements may be either tested individually or combined (composited) and

tested as a unit.

Individuals – conceivable constituent parts of the population.

Gl f t d i li




Laboratory sample – a sample, intended for testing or analysis, prepared fro

a gross sample or otherwise obtained. The laboratory sample must retain th

composition of the gross sample. Often reduction in particle size is necess

in the course of reducing the quantity. Lot – a quantity of bulk material of similar composition whose properties a

under study.

Population – a generic term denoting any finite or infinite collection of

individual things, objects, or events in the broadest concept; an aggregate

determined by some property that distinguishes things that do and do not belong.

Reduction – the process of preparing one or more subsamples from a samp

Sample – a portion of a population or lot. It may consist of an individual or

groups of individuals.

Segment – a specifically demarked portion of a lot, either actual or

hypothetical.

Strata – segments of a lot that may vary with respect to the property under

study.





Subsample – a portion taken from a sample. A laboratory sample may be a

subsample of a gross sample; similarly, a test portion may be a subsample

a laboratory sample.

Test portion (also called specimen, test specimen, test unit, aliquot) – thatquantity of material of proper size for measurement of the property o

interest. Test portions may be taken from the gross sample directly, but oft

preliminary operations such as mixing or further reduction in particle size

necessary.

Types of objects and samples



Types of objects and samples

Object

Homogenous

No change of qualitythroughout the object

Heterogeneous

Discrete change of quality

throughout the object

Continuous change of qual

throughout the object

Homogenous Discrete change Continuous change

• Well-mixed liquid

• Well-mixed gases

• Pure metals

Ore pellets

Tablets

Crystallized rocks

Suspensions

oFluids or gases with gradient

o

Mixture of reacting componeoGranulated materials with

granules much smaller than

sample size



The discrete or continuous changes in object quality can manifest themselv

in time as well as in space.

The composition of the output of a factory changing with time can be seen

a time-related property by sampling the output. It can be seen as a space-

related property by sampling the conveyor belt or the warehouse.

A special type of heterogeneous object exhibits cyclic changes of its

properties. Frequencies of cyclic variations can be a sign of daily influence

as temperature of the environment or shift-to-shift variations. Seasonalfrequencies are common in environmental objects like air or surface water

These several types of heterogeneous objects are seldom found in their

pure state, mixed forms with one or two dominating types are most

common.

The object



The object

“The entity to be described”

An object can be a fertilizer granule as well as a bag or truckload of

fertilizer , or even the quantity produced last week.

As a rule the object is sufficiently described by the four coordinates of

space and time, though in practice other labels may be more useful. Oth

state variable temperature and pressure, for often the sample cannot be

kept in the same condition.

Sample nomenclature



Sample nomenclature

Object/lot/population

Increment Increment

Sample Sample Sample

Gross Sample/Bulk sample

Subsample

(storage)

Subsample

(Laboratory sample)

Laborator

sampleSubsample

(Laboratory sample)

Test portion

The Sample



The Sample“A representative part of the object to be analyzed”

Because representativeness is needed only for the object quality paramete

it can also be stated, “A sample is part of the object selected in such a waythat it possesses the desired properties of the object”

A sample the size of a truckload can be a good sample – that is,

representative of the object- but it cannot be used in the laboratory. A very

small sample can be as good a sample as a larger one, but perhaps it is

impossible to handle.

“The sample must have such dimensions that it can be analyzed.”

The sample must keep the properties the object had at the time of samplingor change its properties in the same way as the object. Thus deterioration

the sample through exposure to the air, the sample container, or

microorganism, for example, should be avoided.

Principles of sampling



Principles of sampling

The purpose of most measurements, and especially chemical measuremen

is to evaluate some property of a material of interest. The value of the

property is then used in a decision process to make judgments on such

questions as its suitability for a specific use, the need for some correctiveaction, or the conformance with some specification.

The samples measured are often the most critical aspect of a measurement

program. They must be relevant and truly represent the universe concern.

In limited cases, the specimen tested is the universe concern. For example

forensic analysis the identification of a single chip of metal could be decis

for settling a legal question.

In most cases, the specimen tested are only a part of the universe of conceThen, it is extremely important to know how the specimen are related to th

universe since the decision process is concerned with the universe and not

with the specimens tested.

Items for consideration when developing a sampling plan



Type of

Sample

Items for consideration when developing a sampling plan

Identification

of population

Chain of

custody

Sample

Number

of sample

Size of

sample

Sampling

SOP

Sampling

calibration

Sample

treatment

containment

Subsampling Storage

holding time

Initial Consideration




The first problem is to identify the universe of concern. There must be a

clear conception of what is to be sampled; hence, the physical, chemical,

and dimensional parameters that define it must be known.

Three kind of universes:

1. Single item, in which the entire universe is evaluated;

2. Discrete lot, consisting of a finite number of discrete individuals, e.g..,

items from a production lot;

3. Bulk, a massive material composed of arbitrary and/or irregular units.A “defined bulk” is one in which the boundaries are clearlydistinguishable while a “diffuse bulk” is one which the limits are ill-defined.

The compositional makeup needs to be considered. Can the universe be

considered to be homogenous? That is to say, can every conceivable sampl

be said to have essentially the same composition, or is the composition

ex ected to var ?





The question of stability of composition needs to be considered in some

cases. The composition of a sample may change once it is removed from it

natural matrix or environment, due to kinetic effect or interaction with a

container, radiation, or air, for example. It may need to be stabilized ormeasured within some safe holding period.

It is necessary to know:

• What substances are to be measured

• What level of precision is required• What compositional information is needed

− Mean composition

− Extremes of composition

− Variability of composition

Purpose of sampling



Purpose of sampling

The sample procedure, or sampling, is a succession of steps performed on

object ensuring that a sample possesses the specified sample quality.

Because the particular sample procedure depends on the purpose of sampl

the various purposes must be delineated.

One purpose of sampling may be collection of a part of the object that is

sufficient for the gross composition of the object, for example, sampling

lots of a manufactured product, lots of raw material, or the mean state of

process. It is desirable to collect a sample that has the minimum size set b

the condition of representative or demanded by handling.

A second purpose of sampling can be the detection: drift or cyclicvariations. Special techniques are required to discover such anomalies as

soon as possible. These also set the condition for sampling frequency and

sample size.



Sampling Plan



p g

There are basically three kinds of sampling plans that can be used in a

measurement process:

1. Intuitive sampling plans may be defined as those based on the judgment of

the sampler. General knowledge of similar materials, past experience, and present information about the universe of concern, ranging from knowledg

to guesses, are used in devising such sampling plans. In the case of

controversy, decisions on acceptance of conflicting conclusions may be ba

on the perceived relative expertise of the those responsible for sampling.

2. Statistical sampling plans are those based on statistical sampling of the

universe of concern and ordinarily can provide the basis for probabilistic

conclusions. Hypothesis testing can be involved, predictions can be made,

and inferences can be drawn. Ordinarily, a relatively large number of samp

will need to be measured if the significance of small apparent differences iof concern. The conclusion drawn from such samples would appear to be

non-controversial, but the validity of statistical model used could be a matt

of controversy.

Sampling Plan



p g

3. Protocol sampling plans may be defined as those specified for decision

purposes in a given situation. Regulation often specify the type, size,

frequency, sampling period, and even location and time of sampling relate

to regulatory decisions. The protocol may be based on statistical or intuiticonsiderations but is indisputable once established. Testing for conforman

with specifications in commercial transactions is another example.

Agreement and definition of what constitutes a valid sample and the meth

of test are essential in many such cases.

When decision are based on identifying relatively large differences,

intuitive samples may be fully adequate. When relatively small

differences are involved and the statistical significance is an issue,

statistical sampling will be required.

When the number of samples required by a statistical sampling plan ma be infeasible, a hybrid plan involving intuitive simplifying assumption

may be used.

Sampling Plan



p g

Whatever kind of sampling plan is developed, it should be written as a protoco

containing procedures (SOPs) that should be followed. It should address:

When, where, and how to collect samples

Sampling equipment, including its maintenance and calibration

Sample containers, including cleaning, addition of stabilizers, and storage

Criteria for acceptance and/or rejection of samples

Criteria for exclusion of foreign objects Sample treatment procedures such as drying, mixing, and handling prior to

measurements

Subsampling procedures

Sample record keeping such as labeling, recording, and auxiliary informat

Chain of custody requirements



Systematic Deviations in Sampling



Some causes of systematic deviations in sampling are the following:

1. The number increments is too small, so that the sample shows a bias. In fac

this is a random deviation.2. The sampling procedure is preferential to one or more object quality

parameters.

3. The sampling procedure causes alterations in the object.

4. The sample changes after the sampling procedure and before the analysis.

5. The sample is altered intentionally.

Sampling parameters



M

G1 G2 G3

A

P

t

x

Lot size P, grab size G, and distance between samples A as a function of time

distance. A sample is the sum of n sample increments, or grabs, taken during

time span G with a frequency 1/A.

Samples for Gross Description



Homogenous objects

If the object to be sampled is homogenous, for example, a well-mixed ta

or lake, it is obvious that here every sample, whatever its size, is a true

copy of the object.

Theoretically one sample suffices. Often in practice more samples are

needed, for instance, to fulfill the requirement that the sample must have

minimum size.

When the sample size too large, such as in precious objets d’art, more

sample are needed.

The size of the gross sample, set by the requirements of the analysis, can

be estimated as follows: assume the sample weight W and the fraction othe component to be estimated is P (1% → P = 0.01; 1 ppm → P = 10-6)

The total amount of component in the sample is T = PW



Heterogeneous objects

The cause of difference between the increment and the object is the

inevitable statistical error in taking an increment from the object. Thecomposition of the grab, the collection of increments that must ultimate

constitute a sample, is given by m and s, with m being the mean

composition of the object and s the standard deviation of this mean.

From statistics it is known that s = ( P A P Bn)1/2 particles or

where s g = standard deviation of the increment P A = fraction of component A in the object

P B = 1 – P A = fraction of component B in the object

n = number of particles.





Assume the same composition as before, 50% A, 50% B, P A = 0.50, and

P B = 0.50. An increment with n = 10 will have composition

m = 50% A;

Two out of three increments will have a composition between 50± 16%or 3 and 7 particles A. The probability that an increment is found outsid

this region is 1 : 3. In fact, the probability of finding an increment with t

composition 10 particles A or B is not negligible: 1% of the increments

will have a composition of 0 or 10 particles of one kind.





If the increment is considered to be representative of the object, when

there is no means to distinguish this increment from another one take

from the object, a method of analysis is required that has an ultimateaccuracy larger than the standard deviation of the increment. When th

accuracy of the analysis is

where sa = standard deviation of the analysis

s A = standard deviation of the method of analysis

N = number of repeated analysis

No distinction can be found between two increments provided s g < sa

(it is known a priori that an increment will not be equal to the object.

This is not important, however, if the difference cannot be seen)





The condition set for an increment to be a sample is s g ≤ sa or

which is equivalent to

It is described by Visman that the sampling variance as the sum of

random and segregation compounds according to

A and B are constants determined by preliminary measurements on the

system. The A term is the random component; its magnitude depends on

the weight w and the number n of sample increments collected, the sizeand variability of the composition of the units in the population. The B

term is the segregation component and depends only on the number of

increments collected and on the heterogeneity of the population.

Samples for Gross Description Heterogeneous objects




Another way to estimate the amount of samples that should be taken

a given increment so as not to exceed a predetermined level of sampl

uncertainty is via the use of Ingamells‟s sampling constant

where W represents the weight of sample; R is the relative standard

deviation (in percent) of the sample composition; and K s is the sampling

constant (Ingamells‟s sampling constant), the weight of sample required

limit the sampling uncertainty of 1% with 68% confidence. The magnitu

of K s may be determined by estimating the standard deviation from a se

of measurements of samples of weight W.




Sampling diagram of sodium-24 in human liver homogenate.

K s

0.1 1 10

1.7

2.1

2.5

2.7

Sample weight, g

C o u n t s , g - 1 . s

- 1 . 1

0 - 2




Gy introduced a shape factor f as the ratio of the average volume of

particles having a maximum linear dimension equal to the mesh size

a sieve to that of a cube which will just pass the same screen:

f = 1.00 for cubes; f = 0.524 for spheres; f ~ 0.5 for most materials.

The particle size distribution factor g is the ratio of the upper size lim

(95% pass screen) to the lower size limit (5% pass screen). For

homogeneous particle size g = 1.00. The composition factor c is

where x is the overall concentration of the component of interest; d xthe density of this component and d g the density of the matrix

(remaining components); c is in the range 5 x 10-5 kg/m3 for highconcentrations of c to 103 for trace concentrations.

Samples for Gross Description Th lib i f l i



The liberation factor l is

where d l is the mean diameter of the component of interest, and d th

diameter of the largest particles. The standard deviation of the samp

s is estimated by

Ingamells related Gy‟s sampling constant to Ingamells‟s constant by

Samples for Gross Description I t ll C l t d Obj t



Objects can be internally correlated in time or space; for example, the

composition of a fluid emerging from a tank does not show random flu

tuations but is correlated in the composition earlier or later sampled product. When the object is internally correlated, no distinct boundaries ex

Factors causing the internal correlation of objects include:

diffusion or mixing within the object, for example, in mixing tanks,

buffer hoppers, and rivers

varying properties of producer of the object, for example, reactors oemitters

When an objects shows a large internal correlation, two adjacent samp

do not differ much from each other. However, the difference between t

samples increases with greater distance. For the case of heterogeneous

objects, the number of increments n depends on the required variance othe sample, this variance being so small that the difference between tw

samples cannot be detected..

Internally Correlated Objects

Samples for Gross Description Internally Correlated Objects




When the object to be analyzed is a process, a stream of material of

infinite length with properties varying in time, the sampling parameters

can be derived from the process parameters.

When the object is a finite part of a process, however, usually called a

lot, the description of the real composition of the lot depends not only

the parameters of the process the lot is derived from but also on the len

of the lot. The sample now must represent the lot, not the process.

Lots derived from Gaussian, stationary, stochastic processes of the firs

order allow a theoretical approach. In practice most lots seem to fulfill

above requirements with sufficient accuracy; an estimate of the mean m

of a lot with size P can be obtained by taking n samples of size G, equaspaced with a distance A; in this case P = nA and the size of the gross

sample S = nG. Here it is assumed that it is not permissible to have

overlapping samples and that A → G.




where

T x = correlation factor of the process

n = number of samples

The only relevant property of the lot here its size p, expressed in unitsThese units may be times units, such as when T x is measured in hours.

this case T x is usually called the time constant of the process. The lot si

is expressed in hours as well. When a river is being described, the lot

can be the mass of water that flows by in 1 day or year.

The correlation factor T x can be called a space constant, however, when

the unit used is of length. Here the lot size is expressed length units.




te a y Co e ated Objects

This is the case, for example, when a lot of manufactured products

contained in a conveyor belt or stored in a pile of material in a warehou

is considered. The values of P, G, and A can be expressed indimensionless units when they are divided by T x.

The correlation factor can also be expressed in dimensionless units, as

when bags of products are produced. In this case the lot size is express

in terms of item as well (number of bags, drums, tablets).

The properties of the sample are increment size g, expressed in the sam

units as T x, and p, the distance between the middle of adjacent increme

a and the number of increments n that form the sample. If the sample

is expressed as a fraction of the object P ( F = G/P ), the relations betwe F and n are depicted in Figure below for a certain value of s x.

1



0.01 0.1 1 10 100 1000 100

0.01

0.1

1

F & n F

Figure The relative sample size F (−) and the relative gross sample size nF (

a function of the relative lot size p for various numbers of sample (s * /s x)

p



Sample uncertainties

T t l i i t f th f th t d t th l d t th i



Total variance consists of the sum of that due to the samples and to their

measurement.

It is the variance of the sample population that is of most concern when

answering questions by measurement. The variance of the samples measur

is related to that of the population and sampling by the expression:

It is good advice to take all the care necessary to make sampling variance

negligible.

Some measurement programs require ancillary data such as pressure,

temperature, flow, and moisture content, to reduce the data to standardconditions. Both random and systematic errors in measurement of these

parameters can reflect similar sources of error in the measurement data.

s2total = s2

sample + s2measurement

s2

sample

= s2

population

+ s2

sampling


St tifi ti hen possible is an insidio s so rce of error in



Stratification

Sample that were initially well-mixed may separate, partially or fully, over

period of time. It may be difficult (perhaps impossible) to reconstitute them

and even the need to do so may not be apparent.

when possible, is an insidious source of error in

analytical sample

Examples: a mixture of solids that may be separate due to gravitational

and/or vibrational forces

emulsions that could demulsify water samples containing suspended matter that later could

plate out on container wall

Whenever stratification is possible, care should be exercised to reconstitute the sample, to

the extent possible, each time a subsample is withdrawn. Otherwise, problems caused by

poor mixing can become even more serious as the ratio of sample increment to residualsample increase.

Any apparent uncompensated uncertainties resulting from segregation in its various aspec

should be considered when evaluating measurement data.


Holding time the maximum period of time that can elapse from samp



Holding time the maximum period of time that can elapse from samp

to measurement before significant deterioration can

expected to occur. When degradation is possible, samp

should be measured before any significant change occur

Max Holding Time

M e a s u r e d

C o n c e n t r a t i o n

C0 Time

Lower Limit

.

If the holding time is considered to be inconveniently short, the conditions ofstorage and/or stabilization must be modified necessitating a new determinatio

of holding time. Very transitory samples may need to be measured immediatel

after sampling.

Statistical Considerations for Sampling

Measurement Situations




Significant Not Significant

A. Measurement Variance X

Sample Variance X

B. Measurement Variance X

Sample Variance X

C. Measurement Variance X

Sample Variance X

D. Measurement Variance X

E. Sample Variance X

Situation A is the most desirable but is seldom encountered. In this case, a

single measurement on a single sample could provide all of the information

required to make a decision. Situation B can be known in advance when the measurement system is in

statistical control, hence a single measurement of a representative sample

be used for decision ur ose.

Statistical Considerations for Sampling





Situation C is more complex in that more than one sample measurement w

be needed in order to relate the data to the universe of interest.

Situation D occurs frequently, especially in environmental analysis. Asampling plan, involving multiple samples, will be needed as in C

Statistical Sampling Plan

Statistics can provide several kinds of information in the development of

sampling plans and the evaluation of measurement data, including:.

The limit of confidence for the measured value of the population mean

The tolerance interval for a given percentage of the individuals in the

population The minimum number of samples required to establish the above interva

with a selected confidence level

Statistical Sampling Plan

S l diti th t t b t h ki t ti ti l ti t



Several conditions that must be met when making statistical estimates:

o The samples must be randomly

selected from the population of

concern

o Each sample must be independent

of any other of sample in the

group

o The type distribution of the

sample must be known, in order

to apply the correct statistical

model.

Could be realized by a samplingoperation that employs a

randomization process and assuranc

that one sample does not influence

another, such as cross-contamination

A Gaussian distribution is often

applicable, or assumed, and such

statistics are easiest to apply.

Basic Assumptions

For statistical planning an estimated or assumed standard deviation is trea



For statistical planning, an estimated or assumed standard deviation is trea

as if it were the population standard deviation.

For statistical calculations, the standard deviation estimate, s, based onmeasurement data, is used in the conventional manner.

In the analysis of variance from several sources, it is assumed that the total

variance is equal to the sum of the various components:

s20 = s2

1 + s22 + ……. + s2

n

In sampling it is used the notation

sA = standard deviation of measurement

ss = standard deviation of samples (within a stratum) sB = standard deviation of between strata

Guidance in Sampling

The minimum number of samples and/or measurements necessary to limit



The minimum number of samples and/or measurements necessary to limit

total uncertainty, to a value E, with a stated level of confidence ( a value z)

For the 95% level of confidence, z = 1.96 ≈ 2

a. Minimum Number of Measurement, nA (ss negligible)

In the above calculation, if nA is more than is considered to be feasible:

• improve the precision of the methodology to decrease sA, or

•

use a more a precise method of measurement if available (smaller sA)• Accept a larger uncertainy.

(Measurement Situation B)




b. Minimum Number of Measurement, ns (sA negligible)

If more samples are required than is feasible:

• use larger sample (smaller ss), or

• use composites (smaller ss), or

• accept a larger uncertainty

(Measurement Situation C)


c When sA and s both are significant



c. When sA and ss both are significant

where nA is the number of measurements per sample.

• The equation above has no unique solution in that several values of

and nA will produce the same value of E.• Compromises will be necessary, taking into consideration the costs o

sampling and of measurement. The nomographs of Provost (Provos

L.P., Statistical Methods in Environmental Sampling for Hazardous

Wastes, ACS Synposium Series 267, ACS, Washington, DC 20036

(1984)) may be helpful in making the various estimates.• The ways to decrease ss and/or sA will decrease the number of

samples and/or measurements required.

(Measurement Situation D)




• When samples come from several strata

Where nB = number of strata sampled

ns = number of samples per strata• No unique solution is possible so that compromises will be required

Cost Consideration

U ll it i t d i li l ith t id ti



Usually, it is necessary to design a sampling plan with cost considerations

mind.

Let

It can be shown that

ns

nA

Cs

CA

C

C

= total number of samples

= number of measurements per sample

= cost per sample of sampling

= cost per measurement

= total cost of program

= ns

Cs

+ ns

nA

CA



Minimum Size of Increments in a Well-Mixed Sample In the case of heterogeneous solids, it is well-known that variability betwee

i i h i i d



increments increases as their size decreases.

It has been shown by Ingamells that the following relation is valid in many

situations.

WR 2 = K s

where W = weight of sample analyzed, g

R = relative standard deviation of sample composition, %K s = sampling constant, required to limit the sampling uncertainty t

1% with 68% confidence.

Once K s is evaluated for a given sample population, the maximum weight,

required for a maximum relative standard deviation of R percent can be

calculated. If the material sampled is well mixed, the relationship holds verwell. For segregated or stratified materials, the calculated value for K s

increases as W increases. This is one way that the degree of mixing can be

judged.

Size of Sample in Segregated Materials

In the case of segregated materials, it has been shown by Visman that the



g g y

variance of sampling, s2s may be expressed by the relation

where A = random component of sampling variance

B = segregation component

W = weight of individual increment, g

n = number of increments

The error due to random distribution may be reduced by increasing W and/

n.

Doubling the number of increments has the same effect as doubling theweight of each increment. However, the error due to segregation is reduced

only by increasing the number of increments sampled.

Size of Sample in Segregated Materials

The constraints A and B must be evaluated experimentally. One way to do



is to collect a set of large increments, each of weight WL, and a set of small

increments, each of weight WS. Each set is measured and the respective

standard deviation, sL and sS are calculated. Then

The degree of segregation

The values for zS > 0.05 can cause serious errors in the estimation of sS if thsingle sampling constant approach is used.

Acceptance Testing Acceptance testing is based on judging whether a lot of material meets

t bli h d ifi ti



preestablished specifications.

• Involving decisions on whether a material tested meets a compositiona

specification such as the carbon content of a cast iron If a representative sample is to be used as the basis for decision, it m

be defined in the plan.

If multiple samples are to be used, then the number, size, and metho

of collection need to be specified.

The permissible variability of material (homogeneity) is oftenimportant and decisions on conformance require statistical sampling

• Consisting in the extent or fraction of defective (out-of-specification)

items in a lot.

The determination of whether the observed percentage of defectivessignificant and exceeds some specified value requires statistical

sampling of the material in question.

Matrix-Related Sampling Problems Sampling Gases

G id d t b i h h th b f



Gases are considered to be microhomogeneous, hence the number of

samples required is related largely differences due to time and location of

sampling. Gases in the atmosphere can be stratified due to emission from point sour

and to differences in density components.

Mixing by diffusion, convection, and mechanical stirring can be more

inefficient than might be imagined. When gas blends are prepared in

cylinders, it can take considerable time for the mixture to equilibrate for tsame reasons

Environmental gas analysis often involves real-time measurement of

samples extracted from the atmosphere. The siting of monitoring stations

a critical aspects of such measurements. The possible degradation of

samples during transit in sampling lines also needs to be considered. Sampling may consist of absorbing some component of interest from an

atmosphere in order to concentrate it to measurable levels or for

convenience of later measurement.

Matrix-Related Sampling Problems Sampling Gases

Gas sampling can consist of collecting a portion in a container The



Gas sampling can consist of collecting a portion in a container. The

possible interaction of components with the walls of the container and al

the possible deterioration and/or interaction of constituents need to be

considered.

Grab samples of gases may be collected by opening a valve of a previou

evacuated container. Composite samples may be obtained by intermitten

opening the valve for predetermined intervals, selected to obtain equal

increments of gas, based on pressure drop considerations.

Gas samples are sometimes collected by condensation in a refrigerated t

One must remember that the composition of gas evolved on warming ca

vary due to differential evaporation unless the entire sample is evaporate

before withdrawal of any part of it. The same problem can be encounter

in withdrawal of the contents of a cylinder gas containing easily condens

components.

Matrix-Related Sampling Problems Sampling Liquids



What has been said about gases applies in principle to the sampling of

liquids.

The concept of holding time is especially applicable to liquids and

especially to aqueous samples.

Liquid samples are often large, to provide the possibility of multiparame

measurement.

Methods of transport need to be considered to minimize breakage and lo

of contents. Sample may appear to be homogenous, but phase separation

can occur as a result of temperature changes, freezing, or long standing.

When such occur, it may be impossible to fully reconstitute the sample.

Matrix-Related Sampling Problems Sampling Solids



The methods used to obtained solid samples depend on whether the

material is massive, consists of aggregates, or is fine granular.

Heterogeneity is a common characteristic, and the statistical sampling

should be considered.

Questions of stability ordinarily are not of major concern, but air oxidati

and/or moisture interactions can cause problems, especially for fine groumaterials with large surface areas.

Matrix-Related Sampling Problems Sampling Multiphases



All of the possible combinations – liquid-solid, liquid-gas, solid-gas, liqu

liquid, solid-solid, and liquid-solid-gas- can be encountered. The samplin

plan applicable will depend on whether a specific analyte, a single phasecomplete analysis, or any variation of the above is of concern.

Once a sample has been obtained and/or removed from its normal

environment, the possibility of phase changes and disruption of phase

equilibria must be considered and addressed as necessary. There can bechanges due to differential volatility, absorption, settling of suspended

matter, demulsification, and related physical and chemical effects that

could change a sample drastically.

In some cases, the phase of interest is separated from its matrix or carrie phase prior to measurement

Principles of MeasurementTerminology



Technique – a scientific principle that has been found useful for providi

compositional information

Method – an adaptation of a technique to a specific measurement

problem

Procedure – the written direction considered to be necessary to utilize a

method

Protocol – a set of definitive instructions that must be followed, withoexception, if the analytical results are to be accepted for a

specific purpose

The hierarchy of methodology: technique – method - procedure - protocol


Absolute Method – method in which characterization is based



entirely on physical (absolute) standards.

Comparative Method – method in which characterization is based o

chemical standards (i.e., comparison with su

standards)

Reference Method – a method of known and demonstrated

accuracy.

Standard Method – a method of known and demonstrated precis

issued by an organization generally recogniz

as competent to do so.

Standard Reference Method – a standard method of demonstrated accura

Definitive Method – a method of known accuracy that has been

accepted to establish the property of some

material (e.g., reference material) and/or to

evaluate the accuracy of other methodology

(term widely used in clinical analysis).




Routine Method – method used in routine measurement of a

measurand. It must be qualified by other

adjectives since the degree reliability is not

implied.

Comparative Method – method in which characterization is based o

chemical standards (i.e., comparison with su

standards)

Field Method – method applicable to nonlaboratory situation

Trace Method – method applicable to ppm range.

Ultra Trace Method – method applicable below trace level.

Macro Method – method requiring more than milligram amou

of sample.

Micro Method – method requiring milligram or smaller amou

of sample.

Table Classes of Methods



Class Precision/Accuracy (P/A) Nomenclature

A < 0.01% Highest P/A

B 0.01 – 0.1% High P/A

C 0.1 – 1% Intermediate P/A

D 1 – 10% Low P/A

E 10 – 35% Semiquantitativ

F > 35% Qualitative

Structure of an Analytical Procedure It consists of six distinguishable steps:



Sampling

Sample preparation

Measuring

Data processing

Testing, controlling, and eventually correcting one or more of

the processing stages

Establishing a quality merit

Structure of an Analytical Procedure No analytical procedure is complete without a proper validation of each o

the stages and of the final result.



g

Based on type of measurements, analytical processes can be divided into

one-dimensional and two-dimensional methods.

Measurement is basically a comparison of an unknown with a known.

Direct comparison, as in determination of mass using an equal-arm

balance

Indirect comparison, as in determination of mass using a spring balanc

Except for the assay of pure substances, the constituent(s) of interest is

usually contained in a matrix that may or may not influence its

measurement. Some measurements can be made in the native matrix with

little or no modification thereof. Emission spectroscopy is an example of

this.

Structure of an Analytical Procedure In others, a modifier may be added to facilitate the measurement or to

provide a matrix common to a variety of measurements and thus minimiz



p y

calibration problems.

Removal of the constituent of interest from its matrix is a common practi

to eliminate matrix problems,

to concentrate the constituent,

to enhance detection limit, or

to increase measurement accuracy

The removal process may utilize dissolution of a sample in a suitable

solvent (or reactant), distillation, extraction, filtration, or chromatographi

separation.

Whenever a removal operation is a part of the measurement process, the

criticality of the steps that are involved needs to be understood thoroughl

and appropriate tolerances must be established and maintained if accurate

results are to be obtained.

Structure of an Analytical Procedure Small modifications of the removal procedure as well as the final

measurement process can constitute a modified if not a different method



measurement with corresponding changes in the accuracy of the output. T

minimize unintentional changes, the development and utilization of an

optimized standard operation procedure (SOP) is highly recommended.

Figures of Merit

Essential Characteristics Desirable Characteristics

Precision Accuracy

Detection level

Sensitivity

Bias

Selectivity Useful range

− Speed− Low cost

− Ruggedness

− Ease of operation

Precision

Precision is one of the most used – and sometimes exclusively used- crite

f lit f l ti l th d



for quality of an analytical method.

Precision expresses the closeness of agreement between repeated testresults. It can be described qualitatively as the quantity that is a measure

the dispersion of results when an analytical procedure is repeated on one

sample. This dispersion of results may be caused by many sources.

It is common practice in describing precision only to imply the sources thcause random fluctuations in the procedure. The analytical procedure

furnishes results that are related to the composition of the sample, so the

scatter of the results will be around the expected value of the result if no

bias exists. This scatter is more often than no of such a nature that it can b

described as a normal distribution.

PrecisionWhen the results are not normally distributed, a simple transformation of

produces a normal distribution. A normal distribution is the frequency



distribution of samples from a population that can be described by a

Gaussian curve.

The normal, or Gaussian, distribution is characterized by the position of t

mean (m) and the half width of the bell-shaped curve at half height,

proportional to the standard deviation (s).

m x

s

Precision



Precision



Precision The harmonic mean is defined by



This is equivalent to a transformation to the inverse of the variables.

An important descriptor of dispersion is the variance or its square root, th

standard deviation. The variance of a population is the mean square

deviation of the individual values from the population mean and is denot

by s 2.

When the variance is estimated from a finite set of data, the symbol var o

are used. The standard deviation s, the positive square root of the varianc

has the same dimension as the data and therefore as a rule is used in the

final result.

Precision



Precision



Accuracy



Accuracy Three mayor components of the measurement process must be attended t

an agreed-on system of measurement



methods of demonstrated accuracy

reference material.

“Accuracy” cannot define a quantity; it only refers to the degree ofattainability of the theoretical concept of “accurate”

An accurate measurement is one that is both free of bias and precise. Bias is the mean of the differences of the results from the known or

assumed true value.

A precise measurement is one that shows no scatter in the results when

repeated.



AccuracyA method with a total error less than 25% is qualified as excellent; 25 – 50% is considered acceptable.



Wolter et.al. defined the maximum total error, the maximum differenc

between a measured value and the true value that can occur with a probability of 95%:

where MTE(u) = upper limit maximum total error

RE(u) = upper limit random error, comprising the c2 distribution

SE(u) or SE(l ) = upper (lower) limit of the systemic error

MTE(u) = RE(u) + max{abs[SE(u)], abs[SE(l )]}



Accuracy Secondary standards.

Measurement by two or more independent and reliable methods whos

estimated inaccuracies are small relative to accuracy required for



estimated inaccuracies are small, relative to accuracy required for

certification.

Measurement via a qualified network of laboratories.

Standard methods.

The composition on the material to be known can be obtained by applyin

an agreed-on method of analysis, for instance, a method issued by the

International Standard Organization (ISO) and the American Society ofTesting Materials (ASTM). The value obtained by analysis according to a

standard method can be assumed to be a “true value‟. This method ofensuring accuracy is often used in trade.

Mean is true. The mean of the results of a number of independent selecte

laboratories can be assumed to be the “true value”

AccuracyWhen the degree of accuracy is estimated, two related questions are often

encountered:

1 When is the bias significant or what is the probability that a found bi



1. When is the bias significant, or what is the probability that a found bi

is real?

2. How can bias be distinguished from random error? (it is dealt with inthe discussions of analysis of variance and sequential analysis).

It should be stressed that random errors may seem to be bias, in particula

when the random errors is large compared to the bias. If a sample that too

small is taken from an inhomogeneous object, the result of the analysis mdeviate appreciably from the results of a thoroughly mixed standard.

Repeating the analysis with a standard method may indicate that no real b

is present. Even when the analytical method shows no bias at all, the

method cannot be designated as accurate when the precision is insufficien

It is useless to look for an accurate method of analysis in situations where

sampling shows a bias or random error that is large compared to bias and

precision of the analysis method

Accuracy



This method is based on the assumption that the mean value of a ser

of observation shows a distribution around the mean value estimatedfrom an infinitely large series, just as individual observations show a

distribution around their mean. The distribution of the means has a

probability density function that resembles the well-known Gaussian

distribution for individual observations. This distribution is called th

distribution.

Accuracy



Limit of Detection Whenever a sample containing a compound in a very low concentration h

to be measured by an analytical procedure, the signal from the measuring

instrument as a rule will be small. It is difficult to decide whether the sign



g

emerges from the component to be determined or from the inevitable noi

produced by the procedure or the instrument.

The uncertainty gives rise to the so-called limit of detection. The quality

this limit can be determined by statistical means.

When one is deciding whether a measured signal originates from themeasured property or does not, some incorrect decisions can be made:

A decision that the component was present in the sample when in fact

was not: error of the first kind.

A decision that the component was not present in the sample when in

it was: error of the second kind.



Limit of Detection



b

a

f( x)

x

The limit of detection is the smallest observed signal ( x) that with a

reliability 1 – a can be considered as being a signal caused by thecomponent to be measured. When the observed signal is smaller than x,

however, it cannot be stated that component is absent.

Limit of Detection



If the signal with magnitude xk is used as a criterion for the presence ofcomponent c, the probability a that an observed signal x > xk caused the

blank is

where P 0( x) = probability distribution of x0 P 1( x) = probability distribution of x1

The probability b that a signal will be observed that is less than xk and

caused by the sample (concentration component x1) is

Limit of Detection



The composition derived from signal x is

where S = sensitivity

Limit of Detection The value of k can be obtained in various ways:

1. When the probability distribution x0 is known with sufficient confide

and proved to be Gaussian, a and k are related as shown in the Table



An often used value of a = 0.0013 (1 – a = 99.87%) results in k = 3, that

Table Value of k

a k

0.5000 0

0.1587 1.0

0.0228 2.0

0.0062 2.50.0026 2.8

0.0013 3.0

0.0005 3.3

Limit of Detection2. When s is not known but estimated a s from finite number of

observations, Student‟s-t must be used (t is correction factor used to

compensate for the uncertainty in the estimation of s)



When for a certain method of analysis the limit of detection x is set equal the mean of the blank signal,

then k = 0 and a = 0.5. This implies that a random result is obtained. The

probability that a signal of a concentration equal to the blank will beconsidered as a signal of the component is 50%. If the obtained signal x =

and k = 3,

b

Limit of Detection



b

a

f( x)

x

The probability a that a signal caused by the blank will be considered as

signal caused by the component is 0.0013 (see the Figure) (assuming

s0 = s1); b, the probability that a signal caused by component will be

considered to be caused by the blank, is now 50%.

Limit of Detection If the mean observed signal x < x, it cannot be stated that c = Sx is smalle

than C , the concentration in the blank, because the reliability is only 0.5.



ba

f( x)

x

Limit of Detection



Limit of Detection



x x

I II III

Three area according Currie:

Limit of Detection



This procedure is applied for enhancing signal-to-noise ratios at the expe

of analysis time

Limit of Detection



Limit of Detection For skewed data a transformation can be obligatory. In gamma spectrome

x-ray spectrometry, and other analytical methods based on particle/radiati

counting, the limit of detection depends more on the counting time than o



the background and blank values.

Sensitivity Sensitivity is the ratio of the quantitative output and the input, for a given

qualitative range. For example S (l = 534 nm) = y/x, where y = extinction

534 nm; x = concentration of component x.



Often, however, y is composed of a part that depends on x and a partindependent of x (blank). As a rule y is not linearly proportional to x over

entire range of possible x and y values.

The range for which S exists and has an unambiguous value is the dynamrange of the procedure.

Mostly for reasons of convenience, it is developed an analytical method i

which S has a constant value in a range as large as possible. The range is

called the linear dynamic range and is expressed in the orders of magnitufor which s can be considered to be constant.

Sensitivity The dynamic range is limited at the lower level by the value of x where y

cannot be distinguished from the noise in y. Here S may have all possible

values.



blank

x

linear dynamic range

Sensitivity In situation where the analytical procedure consists of a chain of procedu

the sensitivity S can be subdivided into sensitivities belonging to the

consecutive operations:



S 1 S 1 S 1 S 1 x x1 x2 x3 xn

y1 y2 y3 yn

y

Cumulative sensitivity for a signal y resulting from a number of consecutiv

operation S i on an input signal x.

Sensitivity The dimension of S depends on the dimension of x and y. If, for example,

is given in grams and y in milliamperes, the sensitivity S is expressed in

mA/g.



Consider the chain of procedures: Concentration (g/L) converted into a potential (mV): S = mV/(g/L)

Potential converted into deflection of a recorder pen (cm): S = cm/mV

Then the ultimate sensitivity is expressed in

The foregoing holds only for static measurements. Under dynamic

circumstances S is time dependent. A sudden disturbance of the value of x

not momentarily followed by an output y; dy/dx is a fuction of t.

Sensitivity If we assume that y is related to x according to a first-order differential

equation, a stepwise disturbance of 0 → x results in



where T x = time constant of procedureBecause the output y is time dependent. There will be a bias between

y(t→∞) = Sx and

x'Sx'

t tm t

y x

Sensitivity If this bias should be limited to a fraction ∆ of the maximum signal, then

Y (t →∞) = (1 - ∆) y

(1 - ∆)Sx = Sx[1 – exp(-T x / t )]



( ) [ p( x )]

and

T x = - t ln ∆

for ∆ = 0.01, T x must be at least 4.6t. However, the measuring instrum

has to return to zero for x → 0, which means that the time lag between sebsequent reading must be at least T x = - 2t ln ∆. This condition imp

that for ∆ = 0.01, T x ≈ 10t .

Many instruments of the type that convert a physical quantity into a curr

or a potential have time constants that range from 100 to 1000ms.

Controllability and Measurability

In studying processes one is interested in an estimation of the quality of

equipment and procedures that are applied for measurement and for proce

control. Criteria that may allow an estimation of this quality are

measurability and controllability



measurability and controllability.

The purpose of control is to diminish the variance of the process. The

parameter that describes the success or the maximum available success of

controlling action is based on diminution of the variance. The controllabi

factor is defined as

A factor r = 1 means that the variations in the process are fully eliminatedand control is 100% successful. A factor r = 0 means that no diminution a

all occurred.

Controllability and Measurability The controllability is determined by several factors, some related to the

control mechanism, others related to the measuring device applied, and al

related to the process to be controlled.



The controllability factor can be divided into two parts: r p is related to control

m is related to measurement

For the purpose of sampling, only m is of interest.

The measurability factor m describes the influence of the measuring devicon control. When the control action is perfect, r p =1; in such a situation it

can be shown that m determines the maximum obtainable value of r :

where m D = e-d (measurability caused by delay time of analysis)

m A = e-a/2

(measurability caused by sampling frequency A-1

)mG = e-g/ 3 (measurability caused by sample size)

m N = 1 – sat e1/2 (measurability caused by precision of analysis)


or

From this equation it follows that the sample size should be kept to a

minimum.



In order to describe the influence of sampling it can be stated that


The controllability is given by the numerical reduction of the value of a

disturbance of the process s x by a controlling action. The remaining value

the disturbance after control, s, and that before define the controllability

factor r according to



g




The frequency of the process fluctuations follows from autocorrelation

function of xt . The autocorrelation gives a relationship between a process

value of time t and that of another time. This predicting element can be us

for measurability and controllability.



When the result of a measurement has been obtained, a prediction can be

made of the value of x in the future, a period t later. The uncertainty in th

prediction is given by the predicted variance




When the accuracy s a of the measuring method and the sample size T g is

included:

Or, for short,

Controllability and MeasurabilityTable Reduction of Disturbances

s min / s x m

0.10 0.995



0.25 0.970.50 0.87

0.87 0.5

In situations encountered in practice it is hardly a disadvantage because lo

m imply (1 – m2

)1/2

≈ 1, so the variance decrease is negligible and thusimmaterial that m may have an approximated value.

Controllability and Measurability Example: the analysis of the nitrogen content of fertilizer. There are many

methods available to do this analysis. These method are compared on the

basis of measurability factors.

The production of fertilizer in a continuous process sets the following



The production of fertilizer in a continuous process sets the following

numerical values:

T a = 30 minutes s x = 1.2% nitrogen

T x = 66 minutes ma = 0.80%

The methods of analysis are listed in the Table below

Table Methods for Analysis of Nitrogen in Fertilizer *

Dead Time SD

Criterion and of Analysis of Analysis

Analytical Techniques (minutes) (%N) (md )a mn m

Total N, classical

distillation 75 0.17 0.32 0.99 0



Total N, DSMAutomated analyzer 12 0.25 0.84 0.97 0

NO3-N, Technicon

Autoanalyzer 15.5 0.51 0.79 0.92 0

NO3-N, specific

electrode 10 0.76 0.86 0.85 0

NH4 NO3/CaCO3 ratio,X-ray diffraction 8 0.8 0.89 0.83 0

Total N, fast neutron

Activation analysis 5 0.17 0.93 0.99 0

Specific gravity, g-ray

absorption 1 0.64 0.98 0.88 0

* A sampling frequency of 2 samples/hour is assumed, which means that ma = 0.80; (md )a =

measurability caused by analysis time; mn = measurability caused by accuracy.


It is seen that the highest m value is given by neutron activation. In a

practical situation the cost of the equipment can play an important role in

solution of a particular procedure. In static situations the analysis time

becomes unimportant.



Here the specific gravity measurement by g-ray absorption, a very inaccu

method, gives the second-best results concerning information and is much

cheaper than neutron-activation method. When it is taken into account tha

the g-ray absorption method can be repeated after 1 minute, (T a = 1 minut

then for this method mtotal = 0.86, by far the best method.

Measuring Functions and Calibration Functions

During the analysis of a sample, a number of actions are performed,

resulting in one or more measured data that are used to calculate quantitat

analytical results.



Analysis may be considered to be a process with input parameters oncomposition, denoted as concentrations xi and with output parameters

measured data yi:

xi PROCEDURE yi

where i = 1,2,3, ……, n j = 1,2,3, …...., m

In order to evaluate from yi the value for xi, the functional relationship

between yi and xi should be known. The establishment of this functional

relationship is called calibration. An analytical method for analysis of n compounds x1, x2,……. xn of a

sample should produce y1, y2,……. yn measured data (m ≥ n).


The functional relationship between measured data y and composition x is

called the calibration function.

In order to apply a given analytical procedure for chemical analysis, each



measurement y j should correspond to a composition xi according to xi = y j ( y j, all x except xi)

This condition implies that in order to measure one concentration value xi, a

other n concentrations x j ( j ≠ i) should be known, or that these values shoul

measured by collecting at least n – 1 more yi values.

For quantitative analysis the solution of m equations with n unknowns (m ≥

x1 = g 1 ( y1, y2,……. ym)

x2 = g 2 ( y1, y2,……. ym)

.

.

xn = g n ( y1, y2,……. ym)




As a rule of thumb, one may state that the resulting n·n matrix should be su

that its determinant has a maximal value. This means that the determinant h

highest possible elements on the diagonal and smallest possible elements on

the nondiagonal positions. A generally accepted method to reduce the numb



of column is to multiply the matrices with their transpose.Example: in a spectrophotometric measurement on chlorine and bromine

dissolved in a suitable solvent, extinction coefficients a x are known. In a mixtu

of Cl2 and Br 2, the measurement results of log (I0/I)l with optical path length 1

cm are as follows:

l (nm) aCl2 (1/mol cm) aBrl2 (1/mol cm) log (I0/I)l = y y

455 4.5 168 0.1995

417 8.4 211 0.2698

385 20 158 0.2980

357 56 30 0.4220333 100 4.7 0.7047

312 71 5.3 0.5023


at ·yl = at ·a·x·d



which become


It is seen that the determinant F is well conditioned because

8214.7 < 18667.81 and < 98659.18.

Application of Cramer‟s rule (provided det ≠ 0) gives



Selectivity and Specificity

In quantitatively analysis the measurement of one particular component o

mixture may be disturbed by other components of the mixture. This mean

that the measurement is nonspecific for the compound under investigatio



A completely specific method for a compound can measure the

concentration of that component, regardless of what other compounds are

present in the sample. The other components do not produce an analytica

signal.

A very important quality criterion of an analytical method is its capabilit

to deliver signals that are free from interferences and give “true results”.The ability to discriminate between the analyte and interfering componen

is expressed as the “selectivity” of a method and measurement system.


A method of analysis is called completely selective when it produces corr

analytical results for various components of a mixture without any mutua

interaction of the components. A selective method thus is composed of a

series of specific measurements.



“Selectivity of a method refers to the extent to which it can determine

particular analyte(s) in a complex mixture without interference from othe

components in the mixture”

The use of the term “selectivity” in analytical chemistry has evolved in

parallel with the development of more sensitive and discriminating methothat have a capability to identify and quantify analytes with less interfere

from other components, similar or dissimilar, than earlier methods were a

to do.

Modern methods are usually designed by combining several measuremen principles introducing their own selectivity to the overall operation. In th

way, very highly selective methods can be obtained.


In situation where the linear system holds,

y = F x

the measurement for component k is specific when only the terms f i



=1,…, n) are non-zero. Here the matrix is called F spec and the value of yfunction of xk only.

For completely selective method each row of matrix F contains only

nonzero element. By interchanging rows, one may get a matrix F , ca

F sel, with all nonzero elements on the diagonal.

In practical situations F sel and F spec contaion other nonzero elements. Th

elements, however, have considerably smaller values than the correspond

f i,k values.

The undesired nonzero elements of F sel and F spec can be used fo

quantitative description of selectivity and specificity.



The Ruggedness Test

Precision involves the determination of the random error under specific

circumstances.

The level at which a method has to be tested depends on its later use.



Whenever an analytical method has to be applied in different site or

laboratories, eventually an interlaboratory test must be carried out to ensu

that the method performs adequately in different circumstances.

The variability of a method under identical circumstances, e.g., in thedeveloper laboratory, is called the repeatability, whereas the precision in

different circumstances is called the reproducibility..

An interlaboratory study can be very costly and should only be undertake

when there is a reasonable chance that a method will be accepted.



The Ruggedness Test

Example of Possible Critical Factors and Levels for HPLC Experiments

Critical Factor Deviation from Nominal Value

Sample preparation factors:

Sample weight 1%



Shake time 20%Sonicate time 20%

Heating temperature 5 0C

Wash volume 30%

Extraction volume 30%

Centrifuge time 20%

Pore size of filter 5 mm⁞

Chromatographic factors

pH 1

Temperature 5 0C

Solvent % 3%

Flow rate 0.1 mL/ minBuffer concentration 1%

Additive concentration 0.5%

The Ruggedness Test

Example of Possible Critical Factors and Levels for HPLC Experiments

Critical Factor Deviation from Nominal Value

Column factors:

Manufacture

B h



Batch⁞

Detector factors

Wavelength 5 nm

Time constant 5

⁞

The Ruggedness Test

The number of factors that have to be tested is usually large. Even when

only the most important factors are selected, a large number of experimen

are necessary. The basic idea is not to change one factor at a time but rath

simultaneously to change several factors in such a way that it is possible

identify the changes that produce an influence on the method‟s performan



Factorial designs are the most suitable for ruggedness testing. They enab

one to establish the influence of each of the factors separately (main effec

and the interactions between them. The number of experiments required i

2

n

, n being the number of factors to be tested. For 7 factors this means 2128 experiments.

In practice full factorial designs can only be used when the number of

factors to be used is low. When it can be assumed that there are no

significant higher order interactions between the factors, which is often arealistic assumption, the fractional factorial designs are most efficient.

Saturated fractional factorial designs enable one to test N – 1 factors with

experiments.

An example of a Placket-Burman design for 12 experiments to test a

i 11 f t Wh th iti l f t id tifi d d th

The Ruggedness Test



maximum 11 factors. When the critical factors are identified and theexperimental design is selected, the experimental work can be carried ou

This involves performing the experiments prescribed by the experimenta

design.

To avoid artifacts, it is necessary to perform these experiment in a randomsequence. The performance of the methods for the different experiments

must be recorded. For HPLC experiments several performance paramete

can be selected. The most relevant of these are the following: the

concentration calculated from the peak height or peak area; the retention

time; the resolution, etc.

Factors

Exp I II III IV V VI VII VIII IX X XI Res

1 76 76 -76 76 76 76 -76 -76 -76 76 -76 7

2 12 12 12 12 12 12 12 12 12 12 12

Placket – Burman Design



Factor

2 12 -12 12 12 12 -12 -12 -12 12 -12 12 3 -21 21 21 21 -21 -21 -21 21 -21 21 21 2

4 25 25 25 -25 -25 -25 25 -25 25 25 -25 2

5 360 360 -360 -360 -360 360 360 360 360 -360 360 3

6 318 -318 -318 -318 318 -318 318 -318 318 318 318 3

7 -29 -29 -29 29 -29 29 -29 29 29 29 29 28 314 314 314 314 314 314 314 314 314 314 314 3

9 -50 50 -50 50 50 -50 50 50 50 -50 -50 5

10 25 -25 25 25 -25 25 25 25 25 -25 -25 2

11 -30 30 30 -30 30 30 30 -30 -30 -30 30 3

12 -230 -230 -230 -230 -230 -230 -230 -230 -230 -230 -230 2

The Ruggedness Test

From the Table it can be seen that for the first factor the experiment 1, 2,

5, 6, and 10 involve the factor at the extreme level, whereas the experime

3, 7, 8, 9, 11, and 12 involve this factor at the nominal level. The main

effect of factor 1 is then calculated as



If no factor has an influence, all the effects should be small and of abothe same size.

If one or two factors have an influence on the performance of the

method, then the values of their main effects will be much larger than

other main effects.

The Ruggedness Test

Obviously a method should not be influenced by the small variations

introduced for the ruggedness test that are expected to be encountered in

practice.

If there are no large difference, the analytical error can be estimated from

h f h i ff



the mean of the main effects:

The estimate of standard deviation reflects the differences that have been

created to simulate the situation that different laboratories have been

carrying out the experiments. If this standard deviation is unsatisfactorily

high, then a interlaboratory test will certainly yield unsatisfactorily result

The Ruggedness Test

If one or two factors show an important influence on the method‟s performance, then the method should be improved to make it more rugge

to the small variations in these factors.

When this is not possible then the method procedure should specify a str



When this is not possible, then the method procedure should specify a strrange these factors, which is chosen so that the performance is adequate

when the factor remain in that interval.

Structure of an expert system for ruggedness analysis

Supervisor

Supervisor

Frame



Factor

Choice

Design

selection

Statistical

results

Chemical

results

Method

improv-

ement

Reselec

factor

levels

Common Data Structure

Explain

Control Chart One of the oldest methods used to monitor and control quality of a

procedure, -in particular, the accuracy and precision- is control chart.

A control chart is a graphic representation on which the values of the

quality characteristic under investigation are plotted sequentially.

l h b i l li d id hi



Control charts are basic tools for quality assurance and provide a graphic

means to demonstrate statistical control, monitor a measurement process

diagnose measurement problem, document measurement uncertainty, and

generally aid in methodology development.

Control charts may also be used to monitor and/or document critical aspe

of samples and sampling operations.

Control Chart



Control Chart Limits are indicated within which the plotted values may be expected to

while the process is in a state of statistical control.

The limits in the X chart are simply “3-sigma” limits, following the origi

Shewart format that indicate the bounds within which substantially all oth d t h ld li h th t i i t t f t ti ti l t l



Shewart format, that indicate the bounds within which substantially all othe data should lie when the system is in a state of statistical control.

An chart is preferable because it is less sensitive to blunders (occasion

wild results), but such charts may not be feasible for the time-consumin

requirements that are the hallmarks of analytical chemistry. Another reasfor preference is that means are more likely to be normally distributed th

individual values, which is important because a normal distribution is

ordinarily assumed when setting control limits.

While an X chart has the advantage of requiring less work, additionalmeasurements may be required on occasion to confirm an apparent

indication of “out of control”.

Control Chart A precision chart consists of standard deviation (for the range which is

related to), evaluated at various times, plotted in a fashion similar to the

above. Ordinarily, several (at least 4) measurements are needed to evalua

the standard deviation on each occasion, and this requirement is its mayo

disadvantage The range chart is a useful type of precision chart



disadvantage. The range chart is a useful type of precision chart.

The maintenance of property charts and precision charts in parallel has

considerable diagnostic value. By inter-comparison of the charts, one can

usually determine whether bias, imprecision, or both are affecting a

measurement process.

Control Limits Control limits can be based on established limits or experimentally

established ones. For manufactured goods, the limits could be based on

acceptability product with the central line as a desired value and the cont

limits as permissible tolerances.

Similarly in measurement the certified value of a control sample could



Similarly, in measurement, the certified value of a control sample couldserve as the central line while the limits could reflect a permissible

uncertainty for the measurement process.

The above approach is no longer considered favorably, since it does not

really monitor either a production process or a measurement process, butmerely indicates what can be gotten away with.

The recommended procedure, almost universally followed today, uses th

means of a number of measured values of the variable as the central line

and the experimentally estimated standard deviation to establish the contand warning limits.



Control Samples The control sample must have a high degree of similarity to the actual

samples analyzed. Otherwise, one cannot draw reliable conclusions the

performance of the measurement system on test samples from its

measurement of control samples.

The control samples must be sufficiently homogenous and stable so the



The control samples must be sufficiently homogenous and stable so theindividual increments measured at various times will have less variability

than of the measurement process.

When the value of the property measured is known with sufficient accur

both the precision of measurement and any systematic measurement erro(bias) can be estimated. Even, if the exact composition is not known, a

suitable control sample can evaluate the stability and precision of the

measurement process.

Ideally, a control chart should bet that is me established for each kind

measurement that is made, and indeed, for each parameter level as well.

Frequency of Control Measurements The required frequency of measurement of control samples will depend o

• the known stability of measurement process

• The importance of the decisions based on the measurement data . Cont

intervals should be chosen with realization that any data obtained durin

the period from last-known-in-control to first-known-out-of-control wi

be in jeopardy



be in jeopardy.

• The cost in time and effort of measuring a control sample. If this is

relatively small, one is well-advised to err on the conservative side and

measure more controls than the minimum number.

Guidance in choosing control sample intervals can be found by applying

“length of run” concept. The interval between retoolings in a production

process may be considered as a production run. Measurement systems m

be studied for possible runs, such as: between calibration, between days,

between shift, between rest periods, between periods of use, betweencritical adjustments.

Range Control Chart

The problem of appropriateness of control samples can be circumvented

using replicate measurements of the actual test samples to monitor

measurement precision.

A special case is the duplicate measurement of a reasonable number o

test samples. It will be remembered that the range of a set of



p gmeasurements (in this case, the difference of the duplicate

measurements) is related to the standard deviation. Accordingly, range

control charts can be developed and used to monitor measurement

precision.

The range control charts do not evaluate systematic departures that woul

affect each measurement in the same way. Thus instrumental shifts and

calibration drifts would go undetected. It be needed to include other cont

samples and/or calibration checks to monitor such problems.

Range Control Chart The average value of the range is calculated from k sets of duplicate

measurements by the expression:

is the central line and the control limits are of using the factors foui h i i l bl h i l l f h f d d



gin the statistical tables. The numerical value for the factor depends on

the number of measurements in the set, and the confidence level for

judgment of control.

Table Factors for use in Duplicate Range Charts and Other Sets of Replicates

Number in Set UWL UCL LCL

2 2.512 3.267 0

3 2.050 2.575 0

4 1.855 2.282 0

5 1.743 2.115 0

6 1.669 2.004 0

Use of Range Control Charts Duplicate measurements may be made of all samples measured, of select

samples in a measurement series of reference materials, or any

combination of the above as designated in the laboratory‟s qualityassurance program.

When the values of R are plotted on the chart, the following courses of

action are recommended:



1. If R is within the warning limit, accept all related data.

2. If R is outside the control limits (UCL), the system is out-of-control.

Reject all data since last-known-to-be-in-control and take corrective

action.

3. If R exceeds the UWL but is within the UCL, accept measurements,tentatively. If the R for the next sample is within the UWL, accept al

the previous data. If it exceeds the UWL, reject all data taken since th

system was last-known-to-be-in control and take corrective actions.

Reestablish control before accepting data.

4. Reestablishment of control should be demonstrated by the results ofthree consecutive sets of measurement that are in control.

Cusum Charts The charts for average and range can be used when sudden changes shou

be detected. The chart is called as the cusum chart for controlling the

quality of analytical methods, whens sudden changes in the method,

application of wrong procedures, interchange of reagents, and the

breakdown of parts of instruments have to be detected.



When more gradual changes should be detected, such as aging of reagen

and obsoleteness of standard curves, another method of plotting the resu

should be used.

The decision is based on the whole of the recent sequence of measureme

Instead of plotting the subgroup averages , a reference value k

chosen and the cumulative sum is calculated:

Cusum Charts Each sums, is obtained from its predecessor by the operation of adding th

new difference; k is a reference value that is chosen equal to the average

but calculation of the cumulative sums (abbreviated to cusum) is simplif

if a rounded value is chosen.

If the mean value of the subgroup averages remains close to the referencl f h diff i i d i h h



value, some of the differences are positive and some negative, so that the

cusum chart is essentially horizontal. However, if the average value of th

process rises to a new constant level, more of the differences become

positive and the mean path slopes upward.

The position of the change in the cusum slope indicates the moment the

change in the quality parameter occurred. However, the visual picture

depends to some extend on the scales chosen for the axes of the chart. If

horizontal distance between the plotted points is regarded as one unit, it i

recommended that the same distance on the vertical scale representapproximately 2s /√n (s is the population standard deviation)

Youden Plot Youden plot method is the round-robin test with graphic reporting that is a

very popular and effective controlling device for analytical laboratories.

In experimental methodology, a round robin test is an interlaboratory te

(measurement, analysis, or experiment) performed independently severa

times. This can involve multiple independent scientists performing the te

with the use of the same method in different equipment, or a variety of



methods and equipment. In reality it is often a combination of the two, f

example if a sample is analysed, or one (or more) of its properties is

measured by different laboratories using different methods, or even just

different units of equipment of identical construction.

A round robin program is a Measurement Systems Analysis techniquewhich uses Analysis of Variance (ANOVA) random effects model to ass

a measurement system.

The Youden plot is a graphical method to analyse inter-laboratory data,

where all laboratories have analysed 2 samples. The plot visualises with

laboratory variability as well as between-laboratory variability

Youden Plot



The original youden plot. PR_A and PR_B

represent similar samples. Notice the two

outliers in the upper right corner of the graph.Circle represents 95% coverage probability

Example of a Youden plot adapted

samples that are not similar. Circle

represents 95% coverage probabilit

A 45-degree reference line is drawn through the Manhattan median

Points that lie near the 45-degree reference line but far from the Manhattan

median, indicate large systematic error.

Points that lie far from the 45-degree line indicate large random error.

Points outside the circle indicate large total error.

Youden Plot Since random errors are equally likely, all four possible results should be

equally likely. Thus in every quadrant formed by subdividing the field

around the mean of the points in the representations space, the number o

points corresponding to the laboratories should be equal.

Otherwise stated, the probability density function consists of concentric



p y y

circles with the mean as center. If the points are considered to have a

normalized Gaussian probability function, the x results are scattered arou

the mean xo with a probability density

The same situation holds for the y results. The standard deviation s of x a

y are the same because the samples were similar; thus

Youden Plot

The pattern formed by the points conclusively demonstrates the major role

played by the various systematic errors. Consider that random errors are re

the cause of the scatter. In such a situation the two determinations, may err

being both low, both high, x low and y high, or x high and y low.



Youden Plot The same statement holds for low results. The elongation of the ellipse is

measure for the accuracy of the method, scales along the x- and y-axes

being the same. The distance of each point along the 45o axis to the mean

a measure of the systematic error that the laboratory is likely to have bee

making.

A laboratory that has large systems0



errors will be represented by a poi

x, y in the Youden plot that sticks t

the 45o axis but the lies quite a

distance from x0

, y0

.

The distance of the point T( x1, y1)

the 45o axis is an estimate of the

random error (the precision) of thi

laboratory.

The distance from S to x0 , y

0 is an

estimate of the systematic error.

y

x

A

B

450

T b

d

S

x0y0

c

st

Youden Plot In fact the Youden plot is a graphic representation of an analysis of varian

The correlation between these methods can be seen by comparing some

parameters.

In an analysis variance the random error s 0 is estimated by



where k = number of laboratory

n = number of analyses for each laboratory (here n = 2)

In the Youden plot (see Figure above)

Youden Plot



In this case n = 2, so

In the analysis of variance the total error

Youden PlotWith n = 2

In the Youden plot,

h i ifi f h diff b l d i i b



The significance of the difference between total error and precision s 0 can b

tested by applying the F -test:

For the value of F greater than the tabulated F (for reliability a, n1 = k-1 and

n2 = k-1) it may be supposed that a systematic error exists:

Youden Plot

The Youden plot can be used not only for comparison of the quality of

laboratories with regard to their results but also for comparison of the

quality of two methods of analysis with regard to their (in)sensitivity to

various laboratories.

In practice often only the qualitative result is presented in graphic form. I

h ld b d h hi k i ll b " li " i



should be noted that this works quite well because "outliers," i.e.,

laboratories having points far from x0 , y0 along the 45o axis, seldom appe

at the same spot when the round-robin test is repeated.

It must be emphasized that quantitative results can be obtained only when

the distribution of the data is not significantly different from a normal one

For a skewed distribution, as may be the case when the samples x and y a

near the limit of detection, an appropriate linearizing methods should be

applied or the quantitative presentation should be abandoned in favor of t

graphic plot.

The Ranking Test

Another way to assess the quality of a laboratory or an analytical method

the application of the ranking test.

The laboratory's score equals the sum of the ranks it receives.



The data from a collaborative test may be arranged in a classification

scheme as shown in Table below. The right-hand side of the table shows t

data substituted by rankings assigned to the laboratories according to the

measured concentration for a series of samples. Rank 1 is given to the

largest amount, rank 2 to the next largest, and so on.

When two laboratories have the same score for the xth place, each

laboratory is assigned the rank x + ½. For a triple tie for the xth place, all

three laboratories get the rank x + 1.

The Ranking Test



The Ranking Test

For n samples, the minimum score is n and the maximum score is nm, whm denotes the number of participating laboratories.

A laboratory that scores the highest amount on every one of the n materia

gets the score of n.

Such a score is obviously associated with a laboratory that consistentlygets high results and the presumption is that this laboratory has a



gets high results, and the presumption is that this laboratory has a

pronounced systematic error. When the results of the laboratories stem

from a random process, the scores tend to cluster around the value n(m

1)/2.

Sequential Analysis

Sequential analysis is a method used to guide statistical interference inobservation series in order to limit the number of observations of that seri

which is not known in advance.

When performing a series of experiments, one sometimes has to decide

whether the experiment gives a result that confirms the hypothesis that is

tested or whether alternatively this hypothesis must be rejected This test



tested or whether, alternatively, this hypothesis must be rejected. This test

can be done by applying statistics to the set of observations.

There are then three possible decisions:

1. From the results obtained it is not yet possible to reject the hypothnor to accept it.

2. The hypothesis is rejected.

3. The hypothesis is acepted.

With the use of the proper statistics, decisions 2 and 3 can be made with

a certain degree of confidence.

Sequential Analysis

If decision 1 is the result, this means that the experiment has to be

continued. The question to be answered is now: how long should the

experiment be continued in order to reach decisions 2 or 3 with sufficient

but not too much- confidence?

By the sequential analysis method a test is conducted that allows a decisio

t b d ft h b ti Th th b b ti i



to be made after each observation. Thus the number observations is

minimal.

Comparison of Sequential and Non-sequential Analyses

Sequential Analysis Non-sequential Analyses

The number of observations required is not The number of observation is known

known in advance. Before the experiment starts

After each observation one out of three After the experiment, comprising a se

possible decisions is selected: of observations, three decisions are1.Continue the experiment and make a new 1. Starts a new series of observations



1.Continue the experiment and make a new 1. Starts a new series of observations

2.Reject the null hypothesis and terminate the 2. Reject the null hypothesis

experiment. 3. Accept the null hypothesis

3.Accept the null hypothesis and terminate the

experiment

After each observation the decision to terminate The experiment is terminated at the e

the experiment may result. of the series of observations.

The decision to terminate the experiment as a

rule results in fewer observations than in non-

sequential analysis.



Sequential Analysis

Mathematical Description

Suppose a hypothesis H has to be tested by performing an experiment

that results in the variable x. Then x1 , x2 , …..xm are m observations of

variable x.

The progress of the test can be shown graphically by computing the

value of a function of all observations, F (m). This value is plotted as a

f nction of the n mber of obser ations



function of the number of observations.

The sample space M m can be subdivided into three mutually exclusive

subspaces:

: space where H is accepted

: space where H is rejected

: space where no decision is possible

Sequential Analysis

Sequential analysis decision areas The direction of these boundar

depends on the values of a and

and the parameters tested (a:

probability of an error of the f

kind -reject H wrongly; b: probability of an error of the

III H0 accepted

R om



f(m)

second kind -accept H wrongly

When the function value cross

one of the boundaries and entezone II or III, the experiment i

terminated by decision dictate

by the zone number.m

II H0 rejected

R m′

I no decision

R m

Data Processing

The first task of an analyst who analyze a process or procedure is to colle

relevant data of measurements on that process or procedure. The

information theory and the sampling strategy should be considered in

assuming what data are relevant and how many measurements

It also important to consider:

the reduction of the data collected according to a certain sampling



the reduction of the data collected according to a certain sampling

strategy without loss of relevant information, and

the methods of data production

Data Processing

There are various methods of data production:

1. A measurement is made and the outcome of the measurement is foun

from the position of a dial on a scale of an instrument. It is the analy

who estimates the position of the dial and converts this to numbers in

digital notation.

2 The measurement is made continuously and translated in the position



2. The measurement is made continuously and translated in the position

a recorder pen on a strip of recorder paper. The analyst reads the

position of the recorded pattern on the time base at times of interest a

converts the selected pen positions to digital numbers.

3. The measurement is made with a fixed frequency, converted

electronically in to a digital notation, and recorded on a device such

disk or magnetic tape. The analyst is not involved in this process, apa

from initiating the procedure of data collection.

Data Processing

The second task, after the numbers are obtained, is to reduce the number measurements, remove the erroneous and irrelevant data, and convert the

measurements in to statements pertaining to the condition of the process o

procedure under control.

In modern analytical procedures, a data acquisition rate of 105

measurements per second is not unusual. In order to convert this enormou



measurements per second is not unusual. In order to convert this enormou

amount of data, collected, for example, during 7 day, into a form that can

handled, data reduction procedures should be applied. Depending on the a

of the analysis, various data reduction procedures may be applied.

Data Processing

Tests for Normality

Since many statistical tests can be applied only when the set of data is

normally (Gaussian) distributed, it is recommended that a test for

normality be made first.

There are several methods to test for normality. An often used test is th

for skewness It signals a symmetry of the distribution Although a pas



for skewness. It signals a symmetry of the distribution. Although a pas

test for skewness does not necessarily prove that the distribution is

normal, a failed test obviously indicates that normal statistics should b

applied with care.A parameter for skewness is the so-called third moment, denoted by m3:

Data Processing

Tests for Normality

An index of skewness that is independent of the unit of measurements

the third moment divided by the cube of the standard deviation. This

parameter is denoted by √b1:



The value of √ b 1 should be zero, but since s normally is not known an

is estimated by s, and because the third moment is also an estimation,some deviation from zero is allowed. Table below gives the probability

various deviations from zero for a normal distribution.



Data Processing

Tests for Normality → Graphic Kolmogorov Test 1. Arrange the data in ascending order.

2. Draw the cumulative frequency line for a normal distribution (G in the

Figure below)



3. Draw two lines parallel to the line of step2, each at a distance D ; D ca

be obtained from Table below

Cumulative frequency for Kolmogorov test

Graphic Kolmogorov Test



4. Draw the cumulative frequency line of the ordered data (S in the Figure

above)

5. If S is found somewhere out side the lines of step3, the distribution cy an

considered not normal with a probability given by the chosen value of Taabove.

Numerical Kolmogorov Test

1. Arrange the data in ascending order.

2. Compute Di = S ( xi) – i/n, with i = sequential data points or frequencies, n

total number of data points, and S ( xi) = corresponding normal cumulative

distribution values.

3. Select the maximum value ( Dm) of Di.



4. If Dm is larger than value given by the Table 4.3, the distribution can be

considered as not normal, with a probability given by the selected a-valu

of Table 3.

An example of both methods is given in the Figure and Table below



Transformation of Data

Data Processing

Occasionally one encounters a distribution that, by one of the tests

mentioned above, departs too far from normality. It would not be safe the

to apply the statistical tests, and it would be a great advantage if, by a sim

transformation of the variable ,an approximately normal distribution coul

be obtained.



A universally applicable transformation for nonnormal data sets to be

created as sets that are as near to normality as possible can be written for

theoretical reason as follows:

Where xi = the measured quantity; = the transformed datum; l = thetransformation parameter; c = constant.

Transformation of Data

Data Processing

For practical applications the equation can be reduced to

The transformation is applied for a range of values of l and each time a tfor normality is performed. The best value of l is chosen.

Th t f ti l ti d b i li d i th ft d



The transformation rules mentioned above are implied in the often used

“intuitive” transformations:

l Transformation

2 Square

1 Unaltered

0.5 Square root

0.333 Cube root

0 log-1 Reciprocal

Information Theory

Analytical chemistry provides optimal strategies for collecting andappreciating irrelevant information on situations and processes in materia

The input into the analytical procedure contains some actual informat

and some latent information (uncertainty).

During the analytical process the actual information increases at theexpense of the latent information.



For example, a sample is analyzed for iron content.

The actual information here is that iron is expected to be present an

the amount is to be determined. The latent information is the unknown quantity of iron.

By an analytical procedure the iron content is measured and found

be 10.82± 0.01% weight.

Thus the uncertainty is decreased from a range of 0 - 100% to that

10.81-10.83% at the expense of the latent information.

Information Theory

The output of analytical chemistry is information that is disclosed by

measurements, the definition of this quantity needs a closer examination.

Information equals diminishing uncertainty, as in an example of the iron

content before analysis equals 0 - 100% and afterward 10.81-10-83%.

General definition of information I equals uncertainty H before - H after analy

Another name for uncertainty H is entropy. The magnitude of H is given b



the number of possibilities (n) before and after the experiment. The most

simple experiment is a decision between two alternatives, yes or no.

The yield of information of such an experiment is selected as a unit of

information, the bit (binary unit). This selection implies

and k = l/logz 2. Thus

ld represents the logarithm with base 2. Other units found in literature are

"nit," based on logarithm with base e, and "dit," based on informationobtained by selecting 1 out of l0 independent possibilities.

Information Theory



If a selection must be made of 1 out of n possibilities with unequal

possibilities, the equation of Shannon should be applied

Information Theory

A possibility to be mentioned is a probability distribution represented by Gaussian function.

Such a distribution may be approximated by means of a histogram with

given width of the classes Dx. The smaller is Dx, the more classes are

needed and the better the approximation.

The probability of occurrence of a given class equals



The entropy of a real continuous distribution may be approximated by entropy of the histogram

Application of Information Theory to Quantitative Analy

The constant Pc of a certain compound in a mixture before and after analyis depicted as follows:

Figure. Probability of occurre

of a concentration P as a funct

of the numerical value of thatconcentration C before (P b) an

ft (P ) l i



Before the analysis each concentration co and ce has an equal probability occurrence; after analysis the concentration is known with a deviation of

Application of the Shannon equation to the situation before analysis yield

after (Pa) analysis


and after analysis



Before analysis

or


After analysis and thus

The information gain I becomes



Application of this procedure in a situation giving a Gaussian distributionthe concentration after analysis implies

in which s is the standard deviation and m the position of the maximum probability.

This distribution results in the following H a value:


and thus

In situations where s is unknown, its value may be estimated frommeasurements and the mean result of the measurement :



A somewhat more generale expression is given by

Here t p,v represents the probability distribution of the analytical results p and

parameter v, representing the number of degrees of freedom.


The gain of information depends on s and n. The increase in the number omeasurements is not proportional to the increase of information: the incre

of information is less than the information expected from the number of

measurements.

The diminishing effect of the estimated number of measurement on

information gain is called redundancy R :



This equation implies that the maximum amount of information to be

gained from n measurements equals n times the information of one sinmeasurement. The interaction between successive measurements thus

supposed to be absent. In the case that correlation exists between

measurements

corr = correlation matrix


The redundancy provides a method for estimation of the mutual interactioof successive measurements. A high value of the redundancy implies that

effects of interruptions otherwise leading to erroneous results are partially

canceled by the correct results in a series of measurements.

In quantitative analysis a calibration graph frequently is used. The questiothen arises: how is the information theory applied here? To begin with, th

standardized function u = ac + b is constructed from m samples with



p

concentrations ci and measurements ui

The analysis of a sample yields a measurement ua and thus a concentratio

co. A set of n parallel measurements result in and .

With a linear dependence of c on u the error in ca, Dca, equals




Application of Information Theory to Qualitative Analys

An accuracy of the melting point measurement corresponds with the widtof the classes in the histogram, the information yield of a melting point

measurement equals

A more accurate measurement of the melting point results in a relativelysmaller reliability interval in comparison with the width of a class and

correspondingly provides more information.



To account for this effect, it is assumed an accuracy for the melting points

equal to k times that of the histogram intervals. The melting pointmeasurement may be considered as a combination of two experiments

1. A class selection procedure with an information yield of

2. Within a class selection of 1 possibility out of a set of k , with an

information yield of I 2 = ld k , or in terms of class broadness D x and

reliability interval D y, I 2 = ld D x/D y


The total amount of information thus becomes

This calculation implies equal probability for all k possibilities within a

given class. In many situations this is not true. In order to deal withsituations where the probability of the situations are not equal, a probabi

distribution function within a class may be defined.



The equations involved in such situations are

for information obtained of intraclass distribution and


For information of intraclass distribution in class i, q j, equals the probabof occurrence of the measured melting point in graph j of class i.

The average intraclass in formation , for the entire histogram equals

which makes the total information



which makes the total information


The equation above refers to the distribution of melting point over a newhistogram with class width m. Based on the histogram and an accuracy ooC in the measurement of the melting points, an amount of information m

be obtained

Based on an estimated number of 106 compounds, the identification of on



particular compound requires an amount of information I = ld n0/n1, whic

here means ld 106/1 = 20 bits.

The amount of information obtained from the melting point, namely, 8 bits, reduces the number of possibilities by 28,5 ≈ 360.

Another useful identification method is thin-layer chromatography

(TLC). Information theory can be applied to this method in order to fin

the best separation and thus the maximal possible identification.


To identify one particular compound out of a set of 8, an amount ofinformation of ld 8 = 3 bits is required.

A theoretical solution for this identification problem is provided by two

separating schemes, one giving 2 bits of information and the other 1 bit.

In practice this solution is not always possible because part of the

information provided by the first solvent may be identical with that

given by the second solvent, which results in a total gain of informatio

of less than 3 bits



of less than 3 bits.

most information is obtained from a combination of two separating

solvents with the highest possible independent information yield.

It is concluded from the RF values that the individual liquids do not provi

enough information for the identification of the compounds a-h (see Tabl

below).




The amount of information is enhanced by application of two successiveseparations with different separating liquids.

There are various combinations of liquids possible.



Can be concluded that the combination of L1 with L3 provides 2 bits of

information only, because the information provided by L3 has been given

already by L1. The same for the combination L2 and L3. The onlycombination L1 with L2 provides all information required for identificatio

Figure Schematic representation of two successive separations with different

separating liquids L1, L2, and L3 for three different selections of liquids.

Filtering

In order to apply measured quantities, in most situations the following

treatment is required:

Removal of nonessential-irrelevant-data (noise)

Reconstruction of essential data

Interpretation of data



Removal of Irrelevant Data (Noise) Noise is the undesired part of a signal.

It is impossible, in practice, to separate noise completely from the rest of

the signal because each noise filter generates noise itself.

A practical yardstick for the applicability of a signal is provided by the

signal-to-noise (SA) ratio.

to measure this ratio one starts by measuring the available power of th

signal wave and of the noise wave



s g w ve d o e o se w ve

In direct current signals the power is proportional to the voltage E; in

alternating current signals it is proportional to the root mean square va

Arms of the waveshape:

with being the average value of the waveshape, and Ai the deviation

the i th measurement from the average

Removal of Irrelevant Data (Noise)

With direct current signals sampled at discrete time intervals the main paof the noise is governed by a variation with time of the signal. The noise

given by the standard deviation of the signal and thus the signal-to-noise

ratio becomes

This equation gives a simple method to calculate S/N from a recorder

signal



signal.

For example, 99% of all observations is found between m - 2.5s and

m + 2.5s (m equals the main value, and s denotes the estimated standarddeviation). Here S/N equals 0.2, the peak-to-peak variance of the signal.

The observation period should be long enough to enable a reliable estima

of S . It is assumed that the signal is distributed around the average value

according to a Gaussian distribution.

Removal of Irrelevant Data (Noise) Other techniques, for example autocorrelation, should be applied for

calculating S/N, where alternating current signal are measured.

Various types of noise that can be recognized by their frequency spectrum

White noise −has a frequency spectrum with a flat shape. It caused by

random processes such as Brownian motion, external influence, andthermal motion of electrons (Johnson noise).

1/f noise −where the power is proportional to 1/ f . Such a situation is

met with for example a drifting signal



met with, for example, a drifting signal.

Red noise −contains more low-frequency noise than the averae value.

Blue noise −contains more high-frequencyn oiset han the averae valu Interference noise −nearly always results from external source with

relatively low chance of occurrence, i.e., switching on or off on powe

undesired components in a spectrum, and so on.

Removal of Irrelevant Data (Noise) A special form of interference noise is impulse noise caused by source

such as the starting up of a powerful engine. All frequencies are met in

this type of noise, making it difficult to eliminate.

Another type of interference noise is deformation noise. Here some

specific frequencies are added to or removed from the signal. Some

possible causes of deformation noise are nonlinearity of the instrumen

transmission losses and reflections.



Frequency spectrum of "white" noise and of "l/ f " noise



Digital Filtering

The equation describes how to calculate a new S , value from an older S t by adding a new measured quantity xt and eliminating the oldest xt-N .

Exponential smoothing requires considerably less storage capacity; becau

nonavailable data are substituted by the mean value: ( xt – xt-N )/ N in previ

equation is replaced by ( xt – S t-1)/ N ′. S t equals the running exponential

mean at time t , and N ' the mean length of the averaging period.Substitution of a : N '-1 yields



the new S t value is obtained from the old by correcting with the scaled

difference between the old value and the most recently measured value.

The scaling factor a is inversely proportional to the length of the runn

interval.

Digital Filtering

Digital signal averaging is a commonly used technique that requires a

many channels as signals to be averaged. Interference noise is notaveraged out by signal averaging.

Th l t d i l b l l di ti i h bl b i i

The advantage of this method over the previous one is that only thmean value S t-1 has to be stored in memory; all measured values ar

accounted for, in contrast to the running average method.



The correlated signals become clearly distinguishable because noise i

eliminated. The correlation time may be used to describe the signal.

Kalman Filtering

Kalman showed that there exists an optimal estimation method that is eve

better than the exponential smoothing method.

A measured quantity yt , obtained at time t , is just a momentary

description of a situation that takes no account of the history of a time

series of measurements.

It is possible to make a better estimate of reconstructed values out of a

time series at time t and to predict the value of new measurements taki

into account the history of that time series. Here the prediction of a



measured value is based on the most recent measurement, and the

expected value is corrected with the values measured in the more dista

past. The weight of the predicting capacity of the measured values fro

the past decreases with increasing distance from the present time. The

application of this technique is called exponential smoothing.

The predicted value for time t , called i, is combined with a measured

value yt , in order to construct a better estimate called* , (eacho f the d

Kalman Filtering

The predicted value for time t , called , is combined with a measured

value yt , in order to construct a better estimate called , (each of the d

xt and yt contains errors, which makes an exact prediction impossible)

Analytical chemistry can be described by a set of very simple equations:

o Here X k, often called the state of the system, can be a concentration



k , y ,

o The equation describes the concentration at time (or sequence) k to

a function of the past; F k is a transition function.o In the case of a stable sample where the concentration will not chan

in time, F k will be 1.

o In the case of a process, F k can for instance be F k = exp(−ak )

o The value of X k will be between certain values we do not know, this

denoted bv W k-1, the systetn error.

Kalman Filtering

o The equation gives the relation between state (e.g., the concentration)

and a measured value X k .

o H k can be the calibration factor, and V k denotes the measurement erro

Now X k can be predicted:



Kalman Filtering

The weight factor K k , determining the relative weights of the present history, can be estimated from

where is the variance in the systern, and Rk is the variance of the

measurement ( Rk = V 2). So a high value of Rk (a high measurementnoise) sets K k to be low and the values from the past get a higher weig

through previous equation.



A larger uncertainty in the system ( P k high) gives more weight to the

measurement. Then can be predicted from

and P k can be estimated from

Kalman Filtering

As this set of equations is recursive a starting value for X k and P k isrequired. These can be guessed from past experience or, when such

experience is lacking, X k can be set to 0, P k to a high value.

As the advantage of the Kalman filter is the exploitation of the

knowledge obtained in the past and updating this knowledgecontinuously, the method can be used to enhance the quality of

analytical measurements



Fourier Convolution

A smoothing filter acts to block high frequencies and allow only low-

frequency signals to pass.

Some examples of smoothing filters are a triangular function ( x = 0, y

1 and x ≥ a, y = 0); the half of a Gaussian function ( x = 0, y = ymax and

x = a, y → 0); and a decreasing exponential function.

The selection of an appropriate smoothing function is a matter of trial

and error. Sometimes the characteristics of the instrument under

consideration are measured under high signal-to-noise conditions.



These characteristics are used afterward in a convolution of signalsmeasured under conditions with high noise levels. Here the spectrum

including noise is convoluted by means of a Fourier transform with a

virtually noise-free signal.

In the frequency domain convolution means multiplication; the place o

time domain asks for a much more elaborate process for convolution.

Here the convolution integral means

T ( y) represents a virtually noise-free convoluted signal: t ( x), the observed

signal: and B( y - x), the smoothing function.

A disadvantage of convolution should be mentioned here: together with t

noise part of the signal, a distortion of the signa loccurs which manifests

Fourier Convolution



itself as a broadening of the peaks (and a lowering of the maximal signal

values because of normalization).

Fourier Convolution

A time series is written as a summation of a number of sine and cosine

terms. This procedure is allowed for each type of function f ( x):

f ( x) = a1 sin x + a2 sin 2 x + a3 sin 3 x + ···· + bo + b1 cos x + ···· + bn cos

In order to calculate the coefficients am, f ( x) is multiplied by sin mx dx (m

1, 2, ...) and integrated between - p and p. This process is visualized asshown in the Figure below.

It is seen that even functions "f(x) = +f(− x)" make all a coefficients zero



It is seen that even functions f ( x) + f ( x) make all a coefficients zero

and odd functions “ f (x) = − f (− x)" make all a coefficients zero.

In situations where the periodicity of sin mx corresponds with a periodici

of f ( x), the product function does not change sign because a change in sig

of sin mx occurs at the same x value where a change in sign of f ( x) occur.



Figure Schematic representation of the convolution of an odd function f ( x

with the sine function sin (4 x).

Deconvolution

Deconvolution offers a method of eliminating distortions of signals that a

caused by the instrument. Here one aims for the recovery of the original

signal, for example, by application of Fourier transform analysis.

As an example the distortion of an optical signal caused by the slit of t

instrument is considered.

The shape of the distorted signal is known to be a broadened line, andorder to recover the original line shape a transformation is required tha

performs the opposite of a convolution calculation:



T ( y) represents the broadened signal, distorted by the Fourier transform

(FT) of the slit width function B( y - x); t ( x) equals the undistorted signal

wants to recover. The aim of the deconvolution process is the recovery of

f ( x).

In the frequency domain deconvolution means division of the FT of the

observed signal by the FT of the distorting function and backtransformatof the resulting FT of the undistorted signal

Deconvolution

The deconvolution process is depicted schematically as follows:

Removal of signal broadening by

deconvolution of the Fourier transform



deconvolution of the Fourier transform

(FT) of the distorted signal by the FT of t

distorting signal followed by the reversetransformation

Deconvolution

The deconvolution results in a smaller line width and thus allows a better

peak separation. However, the deconvoluted signal shows some little extrsignals that are not found in the original spectrum. These signals are

artificial and hamper the detection of low-intensity signals that actually

might be present in the original signal.

In situations where the slit function (or other artificial disturbances due tothe instrument) is unknown it should be measured separately.

A general procedure accepted in spectrometry is to proceed as follows

A spectrum of a component of a mixture is recorded with a narrow slit



p p

width A and another recording is made with identical parameter setting

except for a large slit width B. It is to be noted that recording A gives a better peak separation than recording B. Now the Fourier transformed

spectrum B is divided by the FT of spectrum A in order to produce the

of the slit function C. The back-transformation of C„ yields the slitfunction in the l domain and may be used for transformation purposes

future spectra.

Data Reconstruction and Curve Fitting

The production of measuremnts by one-dimensional methods of analysissuch as melting points, boiling points, indexes of refraction, and so on

mostly do not present problems.

A multiplication or division usually suffices for reconstruction of relev

information.

The situation becomes more complicated for two-dimensional methodanalysis such as are encountered in spectral analysis.

The analysis means measurement of peak positions and of the heights

peak surface.

Diffi l i b d b h f ll i



Difficulties may be caused by the following:

• Noise and drift on the original spectra. Noise may be eliminated by smoothing (the l and the intensity sc

respectively) may be corrected by the application of calibration da

stored in advance.

Baseline detection eliminates drift and the disturbances caused by

functions can be corrected by deconvolution.


• Overlapping peaks. Positions of peaks can be found by differentiating the original

spectrum as a function of time or wavelength.

A stronger overlap asks for more consecutive differentiations in o

to establish peak position.

In practice the number of differentiations is limited because eachdifferentiation enhances the noise.

The peak surface is found by integration over time or wavelength

In order to obtain accurate peak surfaces the baseline should be w

d fi d



defined.

• A baseline mot sufficiently well known.

The baseline is parts of a spectrum where no peaks are found.

A baseline correction is only possible when such parts occur.



In spectroscopy symmetrical peaks are mostly described• by the Lorentz function

• by the Gauss function

• by linear combinations of these functions. AL and A0 are found from

intensity and/or normalization conditions such as




and B represents a relationship with width at half height W: BL = 2/W;

. Nonsymmetric peaks are approximated by means o

empirically adapted standard digital functions. The estimated line shape

can be approximated as accurately as desired by enhancement of the

number of points in the digital function.

Curve Fitting Curve fitting aims for a mathematical description of a series of measured

data by a model with adaptable parameters. In general, the following procedure is applied:

• A mathematical model is chosen on the basis of empirical knowled

on the system under investigation (e.g., Lorentzian line shape or

Gaussian line shape).

• The adaptable parameters in o be determined such as the model sucas line width, peak position, and peak height are calculated in a

manner that makes the sum of the squares of the deviations betwee

calculated and measured data points as small as possible.



p p

• The number of parameters to be found equals the number of peaks

multiplied by the number of parameters per peak.

Suppose the analytical function that is going to represent the measured d

equals

a 1, .., a n are parameters to be determined, i.e., line width and peak positi

y = f (a 1, a 2, a 3, ......., a n, x)

Curve Fitting It is assumed that the number of measured data m is greater than the num

of independent parameters n. The residual sum of squares R that represen

the differences between the calculated values and the measured data Y , is

defined by

and has to be minimized with respect to each a i . The condition



yield n equation of the form



Curve Fitting

Properties of curve fitting by least-squares procedures may be summarize

as follows:

• All data must be stored in a data set before the calculations can be

started.

• The baseline adaptation should be taken into account.

• Partial derivatives with respect to all adaptable parameters should becalculated at each position of the spectrum. Sometimes this calculation

must be repeated at each iteration cycle.

• Matrix manipulations are necessary.

• The starting parameters should be estimated The possible dependence



• The starting parameters should be estimated. The possible dependence

the solution on the starting parameters must always be tested.• Convergence may be accelerated by properly diminishing the paramst

variations.

• The whole procedure may be a failure if an inappropriate mathematica

model is selected.



Mean, Variance, and Covariance

For g ( x) = ax, a representing a constant, this results in

This equation also holds for the continuous probability distribution funct

Apart from the expectation value or mean value of x one may also be

interested in the spread of x values over the series of measurement in ord

to gain insight into the reproducibility of the measurements. The statistic

expression applicable here is the variance of x



expression applicable here is the variance of x,


in the situation with a continuous probability distribution of x,

Another expression used is the standard deviation s ( x):

again selecting the situation E (ax) one may derive



and

where is the absolute value of a


Other useful equations that are easily derived are for z = x + y

Here the covariance between x and y is called cov( x,y)

E ( x) gives the position of the maximum if the x values are represented byGaussian probability distribution,

var( x) is related to the spread of x around the most probable value;

cov( x, y) is related to the relationship between x and y values. (Its

i b i f d f ll d t di )



meaning can be inferred from so-called contour diagrams.)

High values of x are correlated with low values of y

This situation is described by a high negative

covariance between x and y.

Positive correlation with low spread




The magnitude of the covariance can be standardized. The standardized

covariance is called the correlation coefficient r ( x,y):

in which s ( x) = [var( x)]1/2.

A special situation is met when y = ax (a ≥ 0).



Thus


In situations where the value found for x fixes completely

the value to be expected for y, r ( x,y) =± 1.

Where no relationship exists cov( x,y) = 0 and thus r ( x,y) = 0.

In the situations considered so far x as well as y pertain to stationary state

This changes when time-dependent processes are considered. Here the

measured values x vary with time.

The collected values x over a period of time are called a time series.



Time Series The notation of a time series:

Observations are made at times T0, T1, T2,.. .,Tn, giving observed valuZ(T0), Z(T1),..., Z(Tn).

In situations where the time interval between two successive

measurements is a constant h the notation may conveniently be change

into:

For a starting time To = 0 and h equal to the unit of time, Zt , becomes the

measurement at time t .

− measuring time To, To+ h, To + 2h,...,To + nh

− measurements Zo, Z1, Z2,..., Zn



In practice two types of time series may be distinguished: Deterministic time series − in which the future is conrpletely determin

such as Zt = cos (2p ft )

Statistical time series − in which the values expected may be forecast

terms of a probability distribution function

Time Series

A statistical phenomenon that develops in time according to probability

laws is called a stochastic process.



Figure . Example of a stochastic process Z(t) as a function of time with its

probability distribution function P z (t )

Time Series

A stationary process is a special type of stochastic process:

it is a process in statistical equilibrium, which means that the propertieof the process do not depend on the starting time of the observation

period.

Such a process can be subdivided into parts so that all may start at tim

To,.

The subdivision is allowed, provided that a minimal realization lengthobtained or surpassed. Such signals with equal statistical properties ar

called ergodic.

Because the process is statistically determined, each time t provides a

b bilit di t ib ti f Z Thi i li th t ti i f M



probability distribution for Z . This implies that a time series of M

observations should be represented by an M-dimensional vector ( Z 1,

Z 2,..., Z m) with a probability distribution P ( Z 1, Z 2,..., Z m).

Given a stationary process, the condition holds that m measurements Z

Z t 2, ..., Z tm measured at time t 1, t 2,...,t m are the same as the measuremen

Z t 1+t, Z t 2+t, ..., Z tm+t measured at time t 1+t, t 2+,...,t m+t

Time Series

Two important properties of stationary processes are the following:

• The mean value m is a constant given by

• The variance of Z is a constant

In the special situation m = 1 the probability distribution P ( Z t ) is the sam

for all t values and can be denoted as P ( z ).



P ( z ) can be found from a histogram of measurements on time series Z 1, Z Z 3,..., Z n because P ( z ) is a constant for all values of t . The average value m

from a stationary process is found from the average value of the time

series of n → ∞

the variance:

Time Series

For a stationary process the probability of occurrence of a pair of

measurement P ( Z t 1, Z t 2) is a constant for all times t 1 and t 2 provided that ttime interval between t 1 and t 2 is a constant. This is illustrated in Figure

below, where it is apparent that the correlation between Z t +1 and Z t is high

than the between Z t +5 and Z t . In general the correlation decreases upon an

increase of t.



Decrease in correlation with increasing time span of a time series Z(t )

Autocorrelation

The covariance between X t , and Z t+t of a time series can be calculated a

is called the autocovariance g t (auto because one stays within the timeseries):

The collection of g t values for various values of p normalized to thevariance is called the autocorrelation function r t:



Because for a stationary process,

implying



Autocorrelation

A nonstationary process yields j zz (t ) values that do not tend to 0 for high

values. A measured signal that includes drift is always autocorrelated and produces a constant value for j zz (t ).

A periodic function such as a sine function yields an autocorrelation

function with the same periodicity and shape. A series of measurements

suffering from many random high-frequency disturbance can produce avery satisfactory autocorrel;tion function.

The calculation of j zz (t ) may be useful in situations where a series of

process values is hidden by sources of noise



process values is hidden by sources of noise.

Because white noise produces a zero contribution to the autocorrelation

function for t ≠ 0, the influence of random disturbances on theautocorrelation function is eliminated and the calculation of j zz (t ) offers

method of eliminating noise from a time series of measurements.

Autocorrelation



Calculation of Autocorrelation Functions

For a stationary process

and thus



and the autocorrelation function

kimia analitik lanjut

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