On rainflow cycles and the distribution of the number ...

Probabilistic Engineering Mechanics 17 (2002) 123±130

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On rain¯ow cycles and the distribution of the number of interval crossings by a Markov chain PaÈr Johannesson* Mathematical Statistics, Chalmers University of Technology, 412 96 GoÈteborg, Sweden Received 29 November 2000; accepted 13 August 2001

Abstract This paper treats the number of crossings of an interval by a Markov chain observed for ®nite time. The exact marginal distribution is derived, in the form of its probability generating function. A numerical example is presented. Interval crossings have an important application in fatigue of materials. For analysing complex fatigue loads, the so-called rain¯ow cycle counting method is widely used, which is equivalent to counting crossings of intervals. q 2002 Elsevier Science Ltd. All rights reserved. Keywords: Markov chains; Interval crossings; Rain¯ow cycles; Fatigue loads

1. Introduction Let {Xk }1 kˆ0 be a Markov chain with state space {1; ¼; n}; and let NK …i; j† be the number of crossings of the interval ‰i; jŠ by {Xk } in time t ˆ 0; 1; ¼; K: The exact marginal distribution of NK …i; j† is derived, in the form of its probability generating function (PGF). The moment generating function (MGF) is easily obtained from the PGF, and from the MGF, the asymptotic properties of NK …i; j† may be investigated, which results in a normal distribution. The result is directly based on Bartlett [1], who treats the distribution of the frequency of transitions in a Markov sequence. These results are generally not found in textbooks on Markov chains, and is hence not very well known. Some of the results are reviewed in Section 2. 1.1. Rain¯ow cycles and interval crossings When analysing the fatigue life for structures subjected to complex loads, the level crossings have been used for a long time. However, better fatigue life predictions are obtained when using a cycle counting method, which is a rule for pairing local minima and maxima to equivalent load cycles. Especially, the rain¯ow cycle method has been successful, and is now a standard method for engineers, e.g. in the automotive industry. It was invented by Endo in 1967 (see Matsuishi and Endo [14]) as a complicated recursive algorithm. An alternative, but equivalent, de®nition of rain¯ow * Fax: 146-31-772-3508. E-mail address: [email protected] (P. Johannesson).

cycles was given by Rychlik [21]. The principle of this local de®nition is given in Fig. 1. An important feature of the local de®nition is that it makes it possible to establish the equivalence between counting rain¯ow cycles and counting interval crossings. The de®nition in Fig. 1 says; standing on a maximum the load has to reach below level i, in both forward and backward directions, in order to have a rain¯ow minimum less than i. Hence, each cycle with minimum less than i and maximum greater than j gives rise to an upcrossing of the interval ‰i; jŠ; and we have NKrfc …i; j† ˆ NKup …i; j† with NKrfc …i; j† ˆ #

8 > > < > > :

…1† rainflow cycles with

minimum , i and maximum . j for {XK } in time t ˆ 0; 1; ¼; K}

8 9 upcrossing of the closed > > > > < = NKup …i; j† ˆ # interval ‰i; jŠ for {XK } > > > > : ; in time t ˆ 0; 1; ¼; K

9 > > = > > ;

…2†

…3†

where #{´} denotes the number of elements. NKrfc …i; j† is called the rain¯ow counting distribution. A mathematically precise proof of Eq. (1) is given in Rychlik [23, Lemma 7]. The relation (1) is especially important for statistical and mathematical analysis, since it gives an alternative, and often easier, way to analyse the rain¯ow cycle properties of complex loads. The relation has been used in Frendahl

0266-8920/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S 0266-892 0(01)00033-9

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P. Johannesson / Probabilistic Engineering Mechanics 17 (2002) 123±130

Consequently the rain¯ow matrix Frfc and the counting distribution Nrfc contain the same information. The damage according to Eqs. (4) and (5) is easily obtained from the rain¯ow matrix   n X j2i b rfc rfc Db …F † ˆ fij sij with sij ˆ C ; i#j …7† 2 i; jˆ1

Fig. 1. The de®nition of the rain¯ow cycle [21]. From each local maximum Mk one shall try to reach above the same level, in the backward (left) and forward (right) directions, with as small a downward excursion as possible. 1 The minimum, of m2 k and mk ; which represents the smallest deviation from the maximum Mk is de®ned as the corresponding rain¯ow minimum mrfc k : The kth rain¯ow cycle is de®ned as (mrfc k ; Mk ).

and Rychlik [5] and Johannesson [10,11] to obtain explicit formulas for the intensity of interval crossings for Markov models. Further references on computing the intensity of rain¯ow cycles for Markov models are Rychlik [22], Bishop and Sherrat [4], Krenk and Gluver [13], and Olagnon [19]. A law of large numbers for upcrossing and rain¯ow measures is shown in Scheutzow [24] for a very broad class of ergodic processes.

where sij is the damage of a cycle with minimum i and maximum j, and sij ˆ 0 for i . j: Now we will show that the damage can also be expressed directly by using the counting distribution Nrfc . From Eqs. (6) and (7) it can be shown that Db …Frfc † ˆ

Db …Frfc † ˆ

Db …Frfc † ˆ

n X i; jˆ1 n X i; jˆ1 n X i; jˆ1

rfc rfc rfc …nrfc i11; j21 2 ni; j21 2 ni11; j 1 ni; j †si; j

…8†

nrfc i; j …si21; j11 2 si; j11 2 si21; j 1 si; j †

…9†

0 nrfc i; j s i; j

…10†

1.2. Damage

with s 0i; j ˆ si; j 2 si; j11 2 si21; j 1 si21; j11 ; which is exactly the same kind of expression as for fijrfc in Eq. (6).

One often uses a linear damage accumulation hypothesis due to Palmgren [20] and Miner [18]. The fatigue damage caused by a load with stress amplitudes{Sk }; Sk being the amplitude of the kth cycle, is given by: X 1 D…K† ˆ …4† Nsk k

2. Distribution of the number of transitions by a Markov chain

where the sum is extended over all cycles completed at time K. The cycle life NS is obtained from S±N data tests with the constant amplitude S. The principle is that the kth cycle uses a fraction 1=NSk of the total life, and hence fatigue failure occurs when D…K† exceeds one. A commonly used S±N curve is 1 ˆ CSbk with Sk ˆ …Mk 2 mrfc …5† k †=2 NSk where C and b (the damage exponent) are material parameters. For a load with a discrete state space, the rain¯ow cycles can be summarised in the rain¯ow matrix Frfc ˆ … fijrfc †ni; jˆ1; where fijrfc denotes the number of rain¯ow cycles with minimum i and maximum j. The rain¯ow matrix Frfc can be obtained from the rain¯ow counting distribution, denoted n rfc rfc by Nrfc ˆ …nrfc ij †i; jˆ1; with nij ˆ Nk …i; j†; and vice versa, rfc rfc rfc fijrfc ˆ nrfc i11; j21 2 ni; j21 2 ni11; j 1 ni; j

nrfc ij ˆ

iX 21

n X

mˆ1 lˆj 1 1

rfc fml

and …6†

Here we will review results by Bartlett [1] on the distribution of the number of transitions by a Markov chain. The probability generating function (PGF) is given, as well as the moment generating function (MGF) and some asymptotic results. The explicit probability function is given by Whittle [25] and Goodman [8]. Further references on frequency count are Good [7], and also Gabriel [6] for the case n ˆ 2; and Bhat [3] for transient chains. The MGF for a more general model has been treated by Miller [15±17]. Results on related central limit theorems can be found in Keilson and Wishart [12] and HoÈglund [9]. Let {Xk }1 kˆ0 be a Markov chain with transition matrix Q Q ˆ …qij †ni; jˆ1 ;

qij ˆ P…Xk ˆ juXk21 ˆ i†:

…11†

Further, let N ˆ …Nij †ni; jˆ1 ; where the random variable Nij denotes the number of transitions by {Xk }; k ˆ 0; 1; ¼; K; from state i to state j. The PGF for N is de®ned as 2 GN …t† ˆ E4

n Y

i; jˆ1

3 N tij ij 5

ˆ

X n

0 @

n Y

i; jˆ1

1 n tijij P…N

ˆ n†A

…12†

P. Johannesson / Probabilistic Engineering Mechanics 17 (2002) 123±130

where t ˆ …tij †ni; jˆ1 ; tij [ R; and n ˆ …nij †ni; jˆ1 ; nij [ {0; 1; ¼; K}:

Theorem 1. for N is

125

A PGF G…X;Y† …tx ; ty † for the random variables …X; Y† has the following basic properties GX …tx † ˆ G…X;Y† …tx ; 1† and GX1Y …t† ˆ G…X;Y† …t; t†: The following corollary is a simple consequence of these properties.

(Probability generating function) The PGF K

GN …t† ˆ p0 …R…t†† 1

…13†

where

Corollary 1. Let S0,S1,S2,¼,Sm be a disjoint decomposition of the set of indices of N, S ˆ {…i; j† : i ˆ 1; ¼; n; j ˆ 1; ¼; n}: De®ne Nk as X Nij k ˆ 1; ¼; m: Nk ˆ …i;j†[Sk

p0 ˆ …p01 p02 ¼ p0n †

…14†

The PGF G…N1 ;N2 ;¼;Nm † …t1 ; t2 ; ¼; tm † for …N1 ; N2 ; ¼; Nm † is obtained from Theorem 1 by setting ( tk; if …i; j† [ Sk ; k ± 0 tij ˆ 1; if …i; j† [ S0 :

…15†

Next we will give examples of some simple probability generating functions. Consider a MC with 3 states, and transition matrix Q ˆ …qij †3i; jˆ1 ; that is observed at times k ˆ 0; 1; ¼; K: The PGF for the number of visits to the different states (in time k ˆ 1; 2; ¼; K) can be obtained by setting

is the distribution of X0, i.e. p0i ˆ P…X0 ˆ i†; 0

q11 t11

B Bq t B 21 21 B R…t† ˆ B . B . B . @ qn1 tn1

q12 t12 q22 t22 .. . qn2 tn2

¼ q1n t1n

1

C ¼ q2n t2n C C C C; C C ] A ¼ qnn tnn

and 1 is a column vector of ones of length n.

Proof. Here we will outline the proof by Bartlett [1]. For a MC, the expectation of any function FK …X0 ; ¼; XK † may be found if it can be expressed in the product form FK …X0 ; ¼; XK † ˆ

KY 21 kˆ0

f …Xk ; Xk11 †:

…16†

S1 ˆ {…1; 1†; …2; 1†; …3; 1†};

S2 ˆ {…1; 2†; …2; 2†…3; 2†};

S3 ˆ {…1; 3†; …2; 3†; …3; 3†};

S0 ˆ B:

Note that remaining in the same state constitutes new visits. By using Corollary 1 we get the PGF 0 1 q11 t1 q12 t2 q13 t3 K B C C G…N1 ;N2 ;N3 † …t1 ; t2 ; t3 † ˆ p0 B @ q21 t1 q22 t2 q23 t3 A 1:

For by conditioning on X0 ; ¼; XK21 we get

q31 t1

E‰FK …X0 ; ¼; XK †Š ˆ E‰FK21 E‰f …XK21; XK †uXK21 ŠŠ

…17†

and hence it has the same structure as the evolution of the probability distribution of the MC with P…Xk uXk21 † replaced by P…Xk uXk21 †f …Xk21 ; Xk †: Hence, by denoting R ˆ …qij f …i; j††ni; jˆ1 ; the result can be written as: E‰FK …X0 ; ¼; XK †Š ˆ p0 RK 1:

…18†

Now we will show that the PGF, GN …t† ˆ E‰FK …X0 ; ¼; XK †Š; can be calculated as Eq. (18) P by using the product form (16). By writing Nij ˆ K21 kˆ1 1{Xk ˆ i; Xk11 ˆ j} with 1{´} being the indicator function, we obtain: FK …X0 ; ¼; XK † ˆ

KY 21

n Y

kˆ0 i;jˆ1

tij1{Xk ˆi;Xk11 ˆj} ˆ

KY 21 kˆ0

f …Xk ; Xk11 † …19†

and hence f …i; j† ˆ tij : A

q32 t2

q33 t3

The PGF for the number of transitions from state 1 to state 3 is obtained by setting S1 ˆ {…1; 3†}; and S0 ˆ S\S1 ; and again Corollary 1 gives the PGF 0 1 q11 q12 q13 t1 K B C C GN …t1 † ˆ p0 B @ q21 q22 q23 A 1: q31

q32

q33

The MGF for N is de®ned as: 2 0 13 n X MN …t† ˆ E4exp@ tij Nij A5

…20†

and can be rewritten as: 2 3 n Y …etij †Nij 5 ˆ GN …B† MN …t† ˆ E4

…21†

i; jˆ1

i; jˆ1

where B ˆ …etij †ni; jˆ1 : Hence, Theorem 1 and Eq. (21) gives the MGF for N. The moment generating function can be used for investigation of the asymptotic properties of N,

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P. Johannesson / Probabilistic Engineering Mechanics 17 (2002) 123±130

Fig. 2. A sample path where the crossings of the interval ‰i; jŠ are marked with dotted ellipsoids.

see e.g. Bartlett [1]. If Q has no eigenvalues on the unit circle except the simple value l1 ˆ 1; then {Nij } are asymptotically jointly normally distributed. Further, the cumulant generating function KN …t† ˆ log MN …t† has, for small t, the asymptotical form KN …t† , K log l1 …t† where l1 …t† is the eigenvalue such that l1 …0† ˆ 1: Thus, the explicit form of the asymptotic distribution will be determined by the Taylor expansion of log l1 …t† up to the second degree in tij : 3. The distribution of the number of crossings of an interval We will consider crossings of the interval ‰i; jŠ; see Fig. 2. Let {Xk }1 kˆ0 be an ergodic MC with transition matrix Q: Denote by NK …i; j† the number of crossings by {Xk } of the closed interval ‰i; jŠ for k ˆ 0; 1; ¼; K; where X0 has distribution p0 ˆ …p0;i †niˆ1 : The distribution p0 will usually be the stationary distribution p ˆ …pi †niˆ1 of the chain. Our goal is to calculate the distribution of the random variable NK …i; j†: The state space of the Markov chain can be divided into the following three groups: 1. below the interval ‰i; jŠ; S1 ˆ {1; ¼; i 2 1}; 2. in the interval ‰i; jŠ; S2 ˆ {i; ¼; j}; 3. above the interval ‰i; jŠ; S3 ˆ {j 1 1; ¼; n}: The transition matrix can be decomposed according to these

three groups 0 Q11 Q12 B QˆB @ Q21 Q22

Q13

Q31

Q33

Q32

1

C Q23 C A

…22†

where Qml contains transition probabilities of transitions from group m to group l. Fig. 3 describes the transitions by the MC representing up- and down-crossings of the interval ‰i; jŠ; de®ning the following matrices containing transition probabilities ! ! Q11 Q12 0 Q13 R11 ˆ ; R13 ˆ Q21 Q22 0 Q23 R21 ˆ … Q21 R31 ˆ

Q21 Q31

0 †; ! 0 ; 0

R22 ˆ …Q22 †;

R23 ˆ … 0 R33 ˆ

Q23 †

Q22

Q23

Q32

Q33

! :

The MC is in the process of upcrossing ‰i; jŠ if it is in S1, or if it is in S2 and has come from S1. Once the MC is in the process of upcrossing it will remain so when it remains in S1 < S2 ; represented by R11 : When it makes a transition to S3 (from S1 or S2), represented by R13 ; it constitutes an upcrossing. Once the MC has reached S3 it is in the process of downcrossing as long as it remains in S3 < S2 ; represented by R33 ; and a transition to S1, represented by R31 ; constitutes a downcrossing. After reaching S1 the MC is again in the process of upcrossing, until it reaches S3, and so on. At time k ˆ 0 the MC may be in either of the groups S1, S2, and S3. If X0 [ S1 then it is in the process of upcrossing, and if X0 [ S3 then it is in the process of downcrossing.

Fig. 3. Description of the transitions representing up- and down-crossings of the interval ‰i; jŠ:

P. Johannesson / Probabilistic Engineering Mechanics 17 (2002) 123±130

127

If X0 [ S2 then the MC is neither in the process of upcrossing or downcrossing ‰i; jŠ as long as it remains in S2 (represented by R22 †; but when it reaches S1 or S3 it becomes in the process of upcrossing or downcrossing, respectively (represented by R21 and R23 ; respectively).

upcrossings and downcrossings of the interval ‰i; jŠ are Sup ˆ {‰u; 11†; …v; 21†Š; u ˆ 1; ¼; j; v ˆ j 1 1; ¼n};

Theorem 2. (Probability generating function) The PGF for NK …i; j† is

Next, a recursive algorithm for computing the probability P…NK …i; j† ˆ m† from the PGF GNK…i;j† …t† will be derived, by using the same technique as used in Bengtsson and Bondesson [2]. The recursive formulas are well suited for implementation in a matrix manipulation program like Matlab or Maple.

GNK…i; j† …t† ˆ r0 RK 1

…23†

where r0 ˆ … r1

0

r2

0

r3 †

with r1 ˆ …p0;1 ; ¼; p0;i21 †; r2 ˆ …p0;i ; ¼; p0; j †; and r3 ˆ …p0; j11 ; ¼; p0;n † are the probabilities of starting in groups S1, S2 and S3, respectively, and 0 is a vector with j 2 i 1 1 zeros, 0 1 tR13 R11 0 B C C RˆB …24† @ R21 R22 R23 A tR31

0

R33

Sdown ˆ {‰u; 21†; …v; 11†Š;

u ˆ i; ¼; n;

v ˆ 1; ¼; i 2 1};

respectively, see also Fig. 3.

A

Corollary 2. The probability of m crossings by {Xk} of the interval [i,j] in time k ˆ 0; 1; 2; ¼; K is P…NK …i; j† ˆ m† ˆ r0 R…K; m†1

…26†

where R(K,m) is given by the recursion R…k; m† ˆ R0 R…k 2 1; m† 1 R1 R…k 2 1; m 2 1†;

…27†

and 1 is a column vector of ones of length n 1 2… j 2 i† 1 1:

for k ˆ 2; 3; ¼; K and m ˆ 0; ¼; k with starting conditions

Proof. Besides {Xk}, consider a side-information process {Yk} taking values 21, 0, and 11, according to: 8 11 if {Xk } is in process of upcrossing ‰i; jŠ; > > < Yk ˆ 21 if {Xk } is in process of downcrossing ‰i; jŠ; > > : 0 otherwise: …25†

R…1; 0† ˆ R0 ;

(Note that state 0 for {Yk} is a transient state). The process {(Xk ,Yk)} is jointly a MC with state space {…1; 11†; …2; 11†; ¼; … j; 11†; …i; 0†; ¼; … j; 0†; …i; 21†; ¼; …n; 21†}

and with transition matrix given by: 0 1 R13 R11 0 B C C R0 ˆ B @ R21 R22 R23 A R31 Obviously, 8 11 > > < Y0 ˆ 21 > > : 0

0

R33

and using R…k; m† ˆ 0 when m Ó ‰0; kŠ: The matrices R0 and R1 are given by 0 1 0 1 0 R11 0 0 0 R13 B C B C C 0 0 C R1 ˆ B R0 ˆ B @ R21 R22 R23 A; @0 A: 0

if X0 [ S3 ; otherwise …X0 [ S2 †

and hence if X0 has distribution p0 then (X0,Y0) has distribution r0. Now the result follows from Corollary 1 by de®ning the two sets S 01 and S 00 of transitions for {(Xk,Yk)}. Let S 01 ˆ Sup < Sdown denote the set of transitions representing crossings of the interval ‰i; jŠ; and let S 00 denote the other transitions of {…Xk ; Yk †}: The sets of transitions representing

0

R31

R33

0

0 …28†

Proof.

Write R ˆ R0 1 tR1 to get the expansion

RK ˆ …R0 1 tR1 †K ˆ

K X

R…K; m†tm :

…29†

mˆ0

The matrices R(K,m) can be calculated through the recursion in Eq. (27). The recursion can be found by observing that k X

if X0 [ S1 ;

R…1; 1† ˆ R1

mˆ0

ˆ

R…k; m†tm ˆ …R0 1 tR1 †

k X mˆ0

kX 21

R…k 2 1; m†tm

mˆ0

…R0 R…k 2 1; m† 1 R1 R…k 2 1; m 2 1††tm :

By using Theorem 2 and the expansion in Eq. (29), we can write the PGF as: GNK…i;j† …t† ˆ

1 X mˆ0

P…NK …i; j† ˆ m†tm ˆ

1 X mˆ0

r0 R…K; m† 1 tm …30†

128

P. Johannesson / Probabilistic Engineering Mechanics 17 (2002) 123±130

Fig. 4. The distribution of the number of upcrossings of the intervals ‰i; jŠ; i ˆ 2; 3; 4; j ˆ i 2 1; ¼; 3; for K ˆ 10 and K ˆ 100: The means mK …i; j†; marked W, of the distributions are compared with the asymptotic means K mrfc …i; j†; marked p .

where the ®rst expression is the de®nition of the PGF and this completes the proof. A

and make some obvious changes of R0 and R1, since R ˆ R0 1 tR1 ; when using the recursive formulas. 4. Example

Remark 1. (Upcrossings) To get the distribution of the number of upcrossings of the interval ‰i; jŠ we only have to substitute Eq. (24) by: 0

R11

B RˆB @ R21 R31

0 R22 0

tR13

1

C R23 C A R33

…31†

Consider matrix 0 0:1 B B 0:3 B QˆB B B 0:2 @ 0:2

a MC {Xk }1 kˆ0 with 4 states and transition 0:2

0:3

0:1

0:4

0:4

0:1

0:3

0:4

0:4

1

C 0:2 C C C C 0:3 C A 0:1

P. Johannesson / Probabilistic Engineering Mechanics 17 (2002) 123±130

and suppose that the chain starts from its stationary distribution p. By using Corollary 2,with p0 ˆ p; together with the modi®cations described in Remark 1, we will calculate the probability function for NKup …i; j†; denoting the number of upcrossings of the interval ‰i; jŠ in time k ˆ 0; 1; ¼; K: Note that NKup …i; j† is exactly the rain¯ow counting distribution NKrfc …i; j†; see Eq. (1). Numerical calculations of the probability functions for K ˆ 10 and K ˆ 100 are presented in Fig. 4. From the probability function we can compute the mean and the standard deviation of NKup …i; j†; denoted by mK …i; j† and s K …i; j†; respectively. Since {Xk} is stationary and ergodic we know that a:s:

K 21 NKup …i; j† ! mrfc …i; j†;

as K ! 1

…32†

where m rfc …i; j† is called the rain¯ow counting intensity. The derivation of m rfc …i; j† is given in e.g. Johannesson [10,11], where the equivalence Eq. (1) is used to establish that mrfc …i; j† is the probability of starting in S1 and reaching S3, either directly or with intermediate visits to S2. Recall the decomposition of the state space into the groups S1, S2, and S3, and also Eq. (22). In the notation of this paper the formula can be written as:

mrfc …i; j† ˆ p~ …Q13 1 Q12 …I 2 Q22 †21 Q23 † 1

…33†

where p~ ˆ …p1 ; p2 ¼pi21 † and 1 is a column vector of n 2 j 1 1 ones. For ®nite K we have the approximation K 21 mk …i; j† < mrfc …i; j† , mk …i; j† < K mrfc …i; j†:

…34†

In Fig. 4 the means mK …i; j† are compared with the asymptotic means K mrfc …i; j†: Further, the quantities K 21 mK …i; j† are plotted as functions of K, see Fig. 5, and are compared with the asymptotic results mrfc …i; j†: Note that for i ˆ j 2 1;

129

which represents level upcrossings, the two quantities are exactly the same, mrfc …i; j† ˆ K 21 mK …i; j†: According to results by Bartlett [1], see Section 2, we also know that NKup …i; j† will asymptotically be normally distributed, so NKup …i; j† 2 mK …i; j† K 21=2 s K …i; j† should converge in distribution to N(0,1). In Fig. 6 the quantities K 21=2 s K …i; j† are plotted as functions of K. Here we do not have any asymptotic expressions for standard deviations K 21=2 s K …i; j†; however the asymptotic expressions would be possible to compute by using the method sketched in Section 2. For this Markov chain both the normalised mean K 21 mK …i; j† and the normalised standard deviation K 21=2 s K …i; j† are stabilising after about 100±200 observations. 5. Comments concluding Interval crossings for Markov chains have been studied. Results have been given on the exact marginal distribution for the number of crossings, in a ®nite time period, of a given interval. This should be seen as a taste of what is possible to obtain with this technique. The next step would be to compute the bivariate distribution of crossings of two intervals, which would enable us to compute variances and covariances of the interval crossings. This is what is needed to compute the variance of the damage, see Eq. (8) for the computation of the damage from the rain¯ow counting intensity. Acknowledgements I would like to thank Professor Lennart Bondesson,

Fig. 5. The normalised means K 21 mk …i; j†; (Ð), of the number of upcrossings of the intervals ‰i; jŠ; i ˆ 2; 3; 4; j ˆ i 2 1; ¼; 3; as functions of K, compared with the asymptotic means mrfc …i; j†; (- -).

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P. Johannesson / Probabilistic Engineering Mechanics 17 (2002) 123±130

Fig. 6. The normalised standard deviations K 21=2 s K …i; j†; (Ð), of the number of upcrossings of the intervals ‰i; jŠ; i ˆ 2; 3; 4; j ˆ i 2 1; ¼; 3; as functions of K.

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