Christian H Weiß

e w O n l y The INARCH(1) Model for Overdispersed Time Series of Counts

Keywords: ACP model, c control chart, INGARCH model, overdispersion, Poisson-Charlier expansion, simultaneous confidence regions ACP model, c control chart, INGARCH model, overdispersion, Poisson-Charlier expansion, simultaneous confidence regions

The INARCH(1) model for overdispersed time series of counts has a simple structure, a parsimonious parametrization, and a great potential for applications in practice. We analyze two approaches to approximate the marginal process distribution: a Markov chain approach and the Poisson-Charlier expansion. Then approaches for estimating the two model parameters are discussed. We derive explicit expressions for the asymptotic distribution of the maximum likelihood and conditional least squares estimators. They are used for constructing simultaneous confidence regions, the finite-sample performance of which is analyzed in a simulation study. A real-data example from economics illustrates the application of the INARCH(1) model.

Count data processes (X t ) Z , where the X t have state space N 0 = {0, 1, . . .} and where the time indices t are from Z = {. . . , -1, 0, 1, . . .}, are commonly observed in real-world applications, e. g., in insurance (e. g., time series of claim counts), economics (e. g., counts of price changes), statistical process control (e. g., counts of defects), traffic (e. g., counts of accidents), network monitoring (e. g., as a part of an intrusion detection system), epidemiology (e. g., counts of cases of a certain disease) and others; for references, see (Heinen, 2003;[START_REF] Weiß | Thinning operations for modelling time series of counts -A survey[END_REF]. In line with this practical relevance, diverse models for count data time series have been proposed in literature, designed for different types of marginal distribution and autocorrelation structure, see, e. g., the recent reviews by [START_REF] Kedem | Regression models for time series analysis[END_REF][START_REF] Jung | Time series of count data: Modelling, estimation and diagnostics[END_REF][START_REF] Weiß | Thinning operations for modelling time series of counts -A survey[END_REF] propriate type of thinning operation, which are referred to as integer-valued ARMA (INARMA) models. While models based on binomial thinning are commonly used for processes with Poisson marginals, alternative approaches like random coefficient thinning, iterated thinning or quasi-binomial thinning allow to consider marginal distributions with different types of overdispersion [START_REF] Weiß | Thinning operations for modelling time series of counts -A survey[END_REF]. Overdispersion (i. e., a variance greater than the expectation) is commonly observed in practice. Typical reasons for overdispersion are the presence of positive correlation between the monitored events (Friedman, 1993;[START_REF] Poortema | On modelling overdispersion of counts[END_REF][START_REF] Paroli | Poisson hidden Markov models for time series of overdispersed insurance counts[END_REF] or a variation in the probability of the monitored events (Heimann, 1996;[START_REF] Poortema | On modelling overdispersion of counts[END_REF][START_REF] Christensen | Environmental monitoring based on a hierarchical Poisson-Gamma model[END_REF]; further potential causes of overdispersion are discussed by [START_REF] Jackson | All count distributions are not alike[END_REF]). An alternative approach for modeling time series of overdispersed counts are the rather novel INGARCH models, the integer-valued counterpart to the usual generalized autor egressive conditional heteroskedasticity models (for the latter, see [START_REF] Bera | ARCH models: Properties, estimation and testing[END_REF]). The INGARCH models were introduced by (Heinen, 2003) 1 and further investigated by (Ferland et al., 2006;Fokianos et al., 2009;Weiß, 2009;Zhu & Wang, 2009, 2010). The 1 (Heinen, 2003) referred to these models as the autoregressive conditional Poisson (ACP) models. But since these models are closely related to the cardinal GARCH models, (Ferland et al., 2006) suggested to refer to these models as the INGARCH models. INGARCH models are defined by an ARMA-like recursion, and if the range of the model parameters is restricted in an appropriate manner, then a strictly stationary solution of this recursion exists, which also has finite first and second order moments (Ferland et al., 2006). In fact, these models even have an ARMA-like autocorrelation structure, which can be determined from a system of equations being similar to the usual Yule-Walker equations (Weiß, 2009).

In the following, we refer to the INGARCH(p, 0) models as the INARCH(p) models, in analogy to the relation of the usual ARCH and GARCH models, also see (Zhu & Wang, 2009, 2010). These models can also be understood as special generalized linear models (GLM): Using the terminology of Section 1 in [START_REF] Kedem | Regression models for time series analysis[END_REF], the INARCH(p) model is a GLM with Poisson distribution as random component and the identity link as systematic component. These purely autoregressive INARCH(p) models appear to be attractive especially from a practical point of view, since they allow to perform tasks like model identification, model estimation and forecasting relatively easily, also see (Zhu & Wang, 2009, 2010). Also compared to different model families, the INARCH models exhibit several advantages. In contrast to the overdispersed p th order autoregressive models based on a thinning operation, for instance, they only require p + 1 instead of p + 2 parameters, and they have simple transition probabilities as well as a less complex model structure if p > 1. An application of the INARCH models was reported by (Weiß, 2009), who analyzed a time series of monthly claims counts of workers in the heavy manufacturing industry (burn related injuries); this time series was originally presented by (Freeland, 1998). In fact, (Weiß, 2009) showed that already the simple two-parametric INARCH(1) model is able to describe both the observed serial dependence and overdispersion satisfactorily. Also (Zhu & Wang, 2010) for the asymptotic distribution of the conditional least squares (CLS) estimators is derived. It can be used for constructing simultaneous confidence regions, the finite-sample performance of which is investigated in a simulation study. After having illustrated the application of these methods with the real-data example of Section 5, we conclude in Section 6.

The INARCH(1) Model: Definition and

Basic Properties

In this section, we shall briefly review the definition and known basic properties of the INARCH(1) model.

Definition (INARCH(1) Model

) Let (X t ) Z be a process with state space N 0 , let β > 0 and 0 < α < 1. The process (X t ) Z is said to fol- low an INARCH(1) model if X t , conditioned on X t-1 , X t-2 , . . ., is Poisson distributed according to P o(β + α • X t-1 ).
The INARCH(1) model of Definition 2.1 is a stationary Markov chain (Ferland et al., 2006). Since the transition probabilities it is also irreducible and aperiodic and hence ergodic (Zhu & Wang, 2009), also see (Fokianos et al., 2009), who showed ergodicity for the more general INGARCH(1,1) model. All moments exist (Ferland et al., 2006), and its autocorrelation function ρ X (n) := Corr[X t , X t-n ] simply equals α n , like in the standard AR(1) case (Weiß, 2009). It has to be mentioned, however, that the value of α and therefore also of ρ X (n) is restricted to positive values, like in the case of the popular INAR(1) models. The marginal distribution of the INARCH(1) model can be expressed in terms of its cumulants: If

p i|j := P (X t = i | X t-1 = j) = exp (-β -α • j) • (β + α • j) i i! > 0, (1)
µ X (z) := E[exp (z • X t )] denotes the moment generating function (mgf), the cumulant generating function (cgf ) is defined as κ X (z) := ln (µ X (z)). The coefficients κ r of the series expansion κ X (z) = ∑ ∞ r=1 κr r! • z r are referred to
as the cumulants of X, with κ r = κ X (r) (0). The following Proposition 2.2, which was proven by (Weiß, 2009), summarizes essential results about the marginal cumulants of an INARCH(1) process.

Proposition (Marginal

Cumulants) Let (X t ) Z be an INARCH(1)
process according to Definition 2.1. Then the cumulants are determined recursively from

κ 1 = β 1-α , κ n = -(1 -α n ) -1 • ∑ n-1 j=1 s n,j • κ j for n ≥ 2,
s n,0 = 0 and s n,n = 1 for n ≥ 1, s n+1,j = s n,j-1 -n • s n,j
for j = 1, . . . , n and n ≥ 1.

In particular,

κ 1 = β 1-α , κ 2 = β (1-α)(1-α 2 ) , κ 3 = 1+2α 2 1-α 3 •κ 2 , κ 4 = 1+6α 2 +5α 3 +6α 5 (1-α 3 )(1-α 4 ) •κ 2 .
As a consequence of Proposition 2.2, mean µ X , variance σ 2 X , skewness and excess of X t are given by β 1-α ,

β (1-α)(1-α 2 ) , 1+2α 2 1+α+α 2 • √ 1+α β and 1+6α 2 +5α 3 +6α 5 β(1+α+α 2 )(1+α 2 ) ,
respectively. Since σ 2 X > µ X , the INARCH(1) model allows to describe overdispersion.

The Marginal Distribution of INARCH(1)

Processes

An explicit expression for the marginal distribution of an INARCH(1) process (X t ) Z is not known. But since (X t ) Z is an ergodic Markov chain, it follows for the marginal probabilities p i := P (X t = i) that

p i = lim n→∞ p i|j (n) for all i, j ∈ N 0 , ( 2
)
where the n-step transition probabilities

p i|j (n) := P (X t = i | X t-n = j)
with n ∈ N are determined recursively via

p i|j (n) = ∑ ∞ r=0 p i|r • p r|j (n -1). ( 3
p i ≈ p i|j (N ), where p i|j (n) ≈ ∑ M r=0 p i|r • p r|j (n -1) (4)
for arbitrary i, j ∈ N 0 . As an example, one may choose j := ⌈µ X ⌉ (smallest integer not less than µ X ).

This Markov chain approach has the disadvantage that it is computationally difficult and that it requires an appropriate choice of M, N , also see Section 3.1 below. Therefore, it would be desirable to find a more simple approximation of the marginal distribution. Since the marginal distribution is defined through assuming a conditional Poisson distribution, it appears to be a quite natural approach to try to approximate the marginal distribution with the help of the Poisson-Charlier expansion of [START_REF] Barbour | Asymptotic expansions in the Poisson limit theorem[END_REF]

κ (1) = κ 1 , κ (n) = α n • κ n for n ≥ 2.
The proof of Proposition 3.1 is provided by Appendix B.1. In Sections 3.1 and 3.2, we shall investigate the performance of the Poisson-Charlier approximations by comparing the marginal distributions overally and by only considering the effect on the ARL of a c control chart, respectively. We consider approximations up to order 4, see Table A.1, since higher order approximations become too complex for practice and one may apply the Markov chain approach instead.

Study: Approximate Marginal Distribution

To analyze the performance of the first four Poisson-Charlier approximations to the marginal distribution of the INARCH(1) process as given by Table A iteration p (k+1) = Pp (k) as long as the Chebycheff distance between p (k+1) and p (k) was larger than 10 -15 , which determines N . In the following, the main results are reported and illustrated by selected graphs.

Independent of the concrete parametrization, it turned out that the approximations can only be used in the central region of the marginal distribution;

for k → ∞, the relative error of the approximations to the probability p k tends to -100 %, since the approximations decrease faster to 0 than the true probabilities. Otherwise, the performance of the approximations is only moderately influenced by the choice of β, but depends heavily on the choice of α. This is exemplified by the graphs of Figure 1

Study: Approximate ARLs of c Chart

A possible application, where knowledge about the central marginal probabilities of an INARCH(1) process is required, is the computation of average run lengths (ARLs) of control charts. Since such ARLs are computed by averaging about marginal probabilities, it seems possible that the Poisson-Charlier approximations lead to satisfactory results also for larger values of α.

For this purpose, c control charts for different in-control parametrizations and different types of shift in α and β were investigated. A c chart with lower and upper control limits LCL and UCL monitors the observed counts X t directly.

Since an INARCH(1) process is a Markov chain, ARLs can be computed with the well-known Markov chain approach of [START_REF] Brook | An approach to the probability distribution of cusum run length[END_REF], which was a). This result appears to be reasonable in view of the results of Section 3.1. For α 0 = 0.6, at least the approximations of order ≥ 3 can be used, while these approximations lead to errors between -5 % and +5 % for α 0 = 0.75.

Estimation of Parameters

Let x 1 , . . . , x T be a time series from an INARCH(1) process with parameters α, β according to Definition 2.1. In the following, we discuss and compare approaches for estimating these two model parameters, also see (Zhu & Wang, 2009). A related study for the case of an INGARCH(1,1) model was done by (Ferland et al., 2006;Fokianos et al., 2009).

Maximum Likelihood Approach

As already proposed by (Ferland et al., 2006), α and β can be estimated with a conditional maximum likelihood (ML) approach. This type of likelihood

L(β, α) := P (X T = x T , . . . , X 2 = x 2 | X 1 = x 1 ) = ∏ T t=2 e -β-αx t-1 • (β + αx t-1 ) xt /x t ! = e -(T -1)β • e -α ∑ T t=2 x t-1 • ∏ T t=2 (β + αx t-1 ) xt / ( ∏ T t=2 x t !
) .

(5)

The ML estimates are either obtained by numerically maximizing the loga-

rithmic likelihood function ℓ(β, α) := ln ( L(β, α) )
directly, or by solving the following system of equations:

∑ T t=2 x t /( βML + αML x t-1 ) = T -1, βML = 1 T -1 • ( ∑ T t=2 x t -αML • ∑ T t=2 x t-1
) .

(6) (Zhu & Wang, 2009) analyzed the mean square error of the ML estimator in a simulation study.

The observed Fisher information J(β, α), i. e., the negative Hessian of ℓ(β, α),

is given by J(β, α) = ∑ T t=2 J t (β, α) with J t (β, α) =     xt (β+αx t-1 ) 2 xtx t-1 (β+αx t-1 ) 2 xtx t-1 (β+αx t-1 ) 2 xtx 2 t-1 (β+αx t-1 ) 2     . ( 7
)
The proof of formulae ( 6) and ( 7) is provided by Appendix B.2. An explicit expression for the expectation of J t (β, α), the expected Fisher information showed that the ML estimators are asymptotically normally distributed:

I(β, α)
√ T -1 • ( βML -β, αML -α) ⊤ ∼ a N ( 0, I -1 (β, α) ) , ( 8
)
where ∼ a is an abbreviation for asymptotic distribution. In practice, the observed Fisher information J(β, α) can be used to approximate (T -1)•I(β, α), and plugging in the obtained estimates allows, for instance, to approximate the asymptotic standard errors of the ML estimators, also see (Ferland et al., 2006). Related investigations (together with finite-sample results from a simulation study), but concerning the INGARCH(1,1) model, were done by (Fokianos et al., 2009).

Conditional Least Squares Approach

The conditional least squares (CLS) estimators are obtained by minimizing

CSS(β, α) := ∑ T t=2 ( x t -E[X t | X t-1 = x t-1 ] ) 2 = ∑ T t=2 (x t -β -αx t-1 ) 2 . (9)
Evaluating the derivatives of ( 9), one obtains the explicit expression αCLS for the CLS estimators, also see [START_REF] Klimko | On conditional least squares estimation for stochastic processes[END_REF]. The asymptotic distribution of the CLS estimators ( 10) is given by

= ∑ T t=2 X t X t-1 -1 T -1 • ∑ T t=2 X t • ∑ T s=2 X s-1 ∑ T t=2 X 2 t-1 -1 T -1 • ( ∑ T t=2 X t-1 ) 2 , βCLS = 1 T -1 ( ∑ T t=2 X t -αCLS • ∑ T t=2 X t-1 ) (10)
√ T -1 ( βCLS -β, αCLS -α) ⊤ D → N (0, Σ β,α ) with Σ β,α =     β 1-α ( β(1 + α) + 1+2α 4 1+α+α 2 ) -β(1 + α) -(1+2α)α 3 1+α+α 2 -β(1 + α) -(1+2α)α 3 1+α+α 2 (1 -α 2 ) ( 1 + α(1+2α 2 ) β(1+α+α 2 ) )     , (11)

Method of Moments

Since the INARCH(1) process (X t ) Z satisfies E[X t ] = β/(1 -α) and ρ X (1) = α, the method of moments implies to estimate α and β as αMM =

∑ T t=2 (X t -XT )(X t-1 -XT ) ∑ T t=1 (X t -XT ) 2 , βMM = XT • (1 -αMM ), ( 12
)
where XT = 1 T ∑ T t=1 X t denotes the empirical mean of the segment X 1 , . . . , X T .

So both estimators ( 10) and ( 12) are nearly equal to each other. In fact, the arguments of Theorem 3 in (Freeland & McCabe, 2005) carry over, i. e., both estimators have the same asymptotic properties.

Simultaneous Confidence Regions

In The first confidence region is based on the fact that (Y -µ

) ⊤ Σ -1 (Y -µ) is χ 2 p -distributed for a p-dimensional normal vector Y ∼ N (µ, Σ). If z denotes
the γ-quantile of the χ 2 2 -distribution, then we obtain from the asymptotic distribution (11) of the CLS estimators ( 10)

{ (β, α) | ( βCLS -β, αCLS -α) Σ -1 β,α ( βCLS -β, αCLS -α) ⊤ < z T -1 } , ( 13
)
which defines an asymptotically exact simultaneous confidence region for (β, α) on level γ, the CLS 1 confidence region. In complete analogy, we define the ML 1 confidence region based on the ML estimators as

{ (β, α) | ( βML -β, αML -α) J(β, α) ( βML -β, αML -α) ⊤ < z } . ( 14
)
The shape of these regions can be recognized from Figures 5 (a) and (c), respectively, which are computed from the data discussed in Section 5.

Because of computational reasons and the non-rectangular shape of regions ( 13) and ( 14), one might prefer an only approximate but rectangular confidence region for the parameters of the INARCH(1) model. Let z denote the (3 + γ)/4-quantile of the N (0, 1)-distribution, then the region obtained by replacing Σ β,α by Σ βCLS , αCLS and by applying the Bonferroni inequality equals

{ (β, α) | ( βCLS -β) 2 < z 2 T -1 • βCLS( 1+2 α4 CLS + βCLS (1+ αCLS )(1+ αCLS + α2 CLS )) 1-α3 CLS , (α CLS -α) 2 < z 2 T -1 • ( 1 -α2 CLS + αCLS + α3 CLS -2 α5 CLS βCLS (1+ αCLS + α2 CLS )
)} .

( It is referred to as the CLS 2 confidence region, while we refer to are the true process parameters, a simulation study was done. For each combination of the true parameters (β, α), for each γ, T and each type of confidence region, 50,000 time series were generated. From each time series, the respective estimates and confidence region were computed. Then it was checked if this region contained the true parameter tuple. The number of 'successes' divided by 50,000 is an estimate of the true coverage probability.

{ (β, α) | ( βML -β) 2 < z 2 • ( J( βML , αML ) -1 ) 11 , (α ML -α) 2 < z 2 • ( J( βML , αML ) -1 ) 22 )} (
The results are tabulated in Tables 1 to 4.

The results of Table 1 show that the performance of the asymptotically exact CLS 1 region (13) depends heavily on the length T of the available time series and the true autocorrelation parameter α, but only slightly on the particular choice of γ. In general, the difference between γ and the estimated coverage probability (usually < γ) decreases for increasing difference increases for increasing α. In contrast, the ML 1 region ( 14), see

Table 3, appears to be much more robust against T and α, and is therefore clearly preferable especially for small T .

The approximate confidence regions ( 15) and ( 16), see Tables 2 and4, derived from the Bonferroni inequality are always conservative, i. e., their true coverage probability is larger than the nominal level, although this difference decreases for increasing γ. Compared to the asymptotically exact region, the Bonferroni regions lead to an equal or even better performance for large values of γ.

From a practical point of view, the following recommendations can be derived from the Tables 1 and4: The ML 1 region ( 14) appears to be a good choice for any α, γ, T . In contrast, the CLS 1 region (13) leads to a reliable result for a small degree of autocorrelation, say α ≤ 0. In the following, we analyze monthly strike data published by the U.S. Bureau of Labor Statistics (http://www.bls.gov/wsp/). The considered time series of counts describes the number of work stoppages leading to 1000 workers or more being idle in effect in the period. We restrict to the period from January 1994 to December 2002 (108 observations), which was already investigated by [START_REF] Jung | Estimation in Conditional First Order Autoregression with Discrete Support[END_REF]. A line plot of the data is shown in Figure 4 (a), and from the partial autocorrelation function in (c), it becomes clear that a first order autoregressive model seems to be reasonable. [START_REF] Jung | Estimation in Conditional First Order Autoregression with Discrete Support[END_REF] fitted a Poisson INAR(1) model to the data, but since the estimates obtained with different methods deviated heavily from each other, they concluded that such a model is not appropriate. And in fact, see [START_REF] Jung | Estimation in Conditional First Order Autoregression with Discrete Support[END_REF], the data exhibits overdispersion (the empirical variance equals 7.92, being much larger than the mean 4.94), making the Poisson marginal distribution an unreasonable choice, also see Figure 4 (d).

Therefore, we fit an INARCH(1) model to the data, using the approaches discussed in Section 4 before. The estimates obtained with different methods, see

:= x t -Ê[X t | X t-1 = x t-1 ] = x t -β -α • x t-1 for t ≥ 2
shows that the fitted INARCH(1) model seems to be a good choice. 1)). For a description of these models as well as for their parametrization, see [START_REF] Weiß | Thinning operations for modelling time series of counts -A survey[END_REF]. The corresponding ML estimates and asymptotic standard errors (again shown in parentheses and computed from the respective observed Fisher information) are given in Table 5. The performance of these models is measured in terms of the information criteria AIC and BIC, see Table 5, and it becomes clear that the INARCH(1) model clearly performs best.

Finally, we computed the simultaneous confidence regions according to Sec- Tables 1 to 4, where the cells highlighted in gray give the estimated coverages for an INARCH(1) process with true parameters β = 2 and α = 0.6, being close to the situation considered here. It becomes clear that the confidence region CLS 1 has a lower coverage, while ML 1 reaches its nominal level quite well. Also the approximate regions CLS 2 and ML 2 perform quite well for γ ≥ 0.975.

F

A The Poisson-Charlier Approximation

The Poisson-Charlier expansion was introduced by [START_REF] Barbour | Asymptotic expansions in the Poisson limit theorem[END_REF], a good description is provided in Section 4 of [START_REF] Audenaert | Inverse moments of univariate discrete distributions via the Poisson expansion[END_REF]. The Poisson-Charlier expansion is essentially based on the following observation:

If p X (z) := E[z X ] denotes the probability generating function (pgf) of the random count variable X with marginal probabilities p i for i ∈ N 0 and mean µ X , then the factorial cumulant generating function (fcgf ) of X is defined as

k X (z) := ln ( p X (1 + z) ) = ln E[(1 + z) X ] =: ∞ ∑ r=1 κ (r) r! • z r , (A.1)
where the coefficients κ (r) of the series expansion of k X (z) are referred to as the factorial cumulants of X. The factorial cumulants are related to the usual cumulants through the relation [START_REF] Douglas | Analysis with standard contagious distributions[END_REF]p. 470) If now, in turn, the fcgf of X is known, then its pgf is obtained as .3) This leads to the idea to approximate the true pgf of X by the m th order approximation .4) where in the case of the Poisson distribution, the first order approximation already

κ (n) = ∑ n j=1 s n,j • κ j , (A.2)
p X (z) = exp ( k X (z -1) ) = exp ( ∑ ∞ r=1 κ (r) r! • (z -1) r ) . ( A
p X (z) ≈ exp ( ∑ m r=1 κ (r) r! • (z -1) r ) , ( A
gives the exact pgf.

The Poisson-Charlier expansion of [START_REF] Barbour | Asymptotic expansions in the Poisson limit theorem[END_REF] is a further refinement of this approach. Let π i := e -κ 1 • κ i 1 /i! for i ∈ N 0 denote the Poisson probabilities of the P o(κ 1 )-distribution, where κ 1 = µ X = κ (1) , and let ∇ denote the (backward) difference operator, i. e., ∇π i = π i -π i-1 . Then the m th order Poisson-Charlier approximation of the true probability p i is given by f m (∇) • π i , where f m is defined as the (m -1) th order Taylor polynomial in z around z = 0 and evaluated in z = 1 of the function

f (z, ∇) := exp ( 1 z • ∞ ∑ r=2 κ (r) r! • (-z∇) r
) , (A.5) see [START_REF] Audenaert | Inverse moments of univariate discrete distributions via the Poisson expansion[END_REF]. The first four approximations are summarized in

f 1 (∇) = 1, f 2 (∇) = 1 + 1 2 κ (2) ∇ 2 , f 3 (∇) = 1 + 1 2 κ (2) ∇ 2 -1 6 κ (3) ∇ 3 + 1 8 κ 2 (2) ∇ 4 , f 4 (∇) = 1 + 1 2 κ (2) ∇ 2 -1 6 κ (3) ∇ 3 + ( κ 2 (2) 8 + κ (4) 24 ) ∇ 4 -1 12 κ (2) κ (3) ∇ 5 + 1 48 κ 3 (2) ∇ 6 .
belonging to the P o(µ X )-distribution. The higher order approximations account for the deviation of the true distribution of X from the Poisson distribution, where the deviation is measured in terms of the factorial cumulants.

B Proofs

(z) = exp ( β(z -1) ) • p X ( exp ( α(z -1)
) ) , also see (Weiß, 2009). Hence, we obtain for the fcgf k X (z) the relation

k X (z) = βz + κ X (αz).
Differentiating this relation, it follows that

k ′ X (z) = β + α • κ ′ X (αz), and
k (n) X (z) = α n • κ (n) X (αz) for n ≥ 2.

B.3 Proof of Formula (11)

The INARCH(1) process according to Definition 2.1 is an ergodic stationary Markov chain, having the moments [START_REF] Douglas | Analysis with standard contagious distributions[END_REF]. Therefore, the asymptotic distribution of the CLS estimators (10) can be determined by applying the results of Section 3 in [START_REF] Klimko | On conditional least squares estimation for stochastic processes[END_REF]) (alternatively, Theorem 4 in (Zhu & Wang, 2009) might be applied). Adapting their notations, we define

µ X = κ 1 = β 1-α , µ X,2 := E[X 2 t ] = κ 2 + µ 2 X = β(1+β(1+α)) (1-α)(1-α 2 ) , µ X,3 := E[X 3 t ] = κ 3 + 3κ 2 µ X + µ 3 X = β (1-α) 3 • ( 1+2α 2 (
g t-1 (β, α) := E[X t | X t-1 , . . .] = β + α • X t-1 , u t (β, α) := X t -g t-1 (β, α) = X t -β -α • X t-1 .
(B.5)

The function g t-1 (β, α) is easily shown to satisfy the regularity conditions (i) to (iii) defined on page 634 of [START_REF] Klimko | On conditional least squares estimation for stochastic processes[END_REF]. Especially, we obtain that

V :=     E [( ∂ ∂β g t-1 ) • ( ∂ ∂β g t-1 )] E [( ∂ ∂β g t-1 ) • ( ∂ ∂α g t-1
)] (B.6) see formula (B.4). The inverse of V equals

E [( ∂ ∂α g t-1 ) • ( ∂ ∂β g t-1 )] E [( ∂ ∂α g t-1 ) • ( ∂ ∂α g t-1 )]     =     E[1] E [ X t-1 ] E [ X t-1 ] E [ X 2 t-1 ]     =     1 β 1-α β 1-α β(1+β(1+α)) (1-α)(1-α 2 )     ,
V -1 =     1 + β(1 + α) -(1 -α 2 ) -(1 -α 2 ) (1-α)(1-α 2 ) β     . (B.
) • u 2 t ]     =     E [ 1 • (β + α • X t-1 ) ] E [ X t-1 • (β + α • X t-1 ) ] E [ X t-1 • (β + α • X t-1 ) ] E [ X 2 t-1 • (β + α • X t-1 ) ]     = β 1-α •     1 α+β(1+α) 1-α 2 α+β(1+α) 1-α 2 β(1+2α) (1-α)(1-α 2 ) + β 2 (1-α) 2 + α(1+2α 2 ) (1-α 2 )(1-α 3 )     .
(B.8) Formula (B.8) follows, since

E [ f (X t-1 ) • u 2 t (β, α) ] = E [ f (X t-1 ) • E[(X t -β -α • X t-1 ) 2 | X t-1 ] ] = E [ f (X t-1 ) • V [X t -β -α • X t-1 | X t-1 ] ] = E [ f (X t-1 ) • (β + α • X t-1 ) ]
because of the conditional Poisson distribution. So we can apply Theorem 3.2 of [START_REF] Klimko | On conditional least squares estimation for stochastic processes[END_REF] to obtain the asymptotic distribution of the CLS estimators (10):

√ T -1 ( βCLS -β, αCLS -α) ⊤ D → N (0, V -1 WV -1 )
, where

V -1 WV -1 =     β 1-α ( β(1 + α) + 1+2α 4 1+α+α 2 ) -β(1 + α) -(1+2α)α 3 1+α+α 2 -β(1 + α) -(1+2α)α 3 1+α+α 2 (1 -α 2 ) ( 1 + α(

  . Particularly popular are ARMA-type models based on an ap-

  proposed to use an INARCH(1) model for a time series of download counts. A further real-data example, where again an INARCH(1) model proves to be appropriate, is presented in Section 5 below: monthly strike data published by the U.S. Bureau of Labor Statistics. So in spite of its simple structure and its parsimonious parametrization, the INARCH(1) model seems to be of great practical relevance. It can be considered as a counterpart to the very popular Poisson INAR(1) model, but being able to describe overdispersion. For these reasons, we shall analyze the INARCH(1) model in more details in this article. In Section 2, we review the definition and basic properties of the INARCH(1) model. Section 3 shows how to approximate the marginal process distribution with the help of the Poisson-Charlier expansion. The goodness of this approximation is investigated, also in view of approximating the average run 5 ) of a c control chart. Section 4 discusses approaches for the estimation of the two model parameters. In particular, an explicit expression

Figure 1 :Figure 2 :

 12 Figure 1: Marginal distribution (P) and relative errors of Poisson-Charlier approximations (PCn) for (a) (β, α) = (4.5, 0.1), (b) (β, α) = (3.75, 0.25), (c) (β, α) = (3, 0.4), (d) (β, α) = (2.5, 0.5), (e) (β, α) = (2, 0.6).

βα

  Figure 3: ARL(β) of c chart and relative errors of Poisson-Charlier approximations (PCn) for (a) (β 0 , α 0 ) = (2.5, 0.5), (b) (β 0 , α 0 ) = (2, 0.6), (c) (β 0 , α 0 ) = (1.25, 0.75).

→

  where D abbreviates convergence in distribution. The proof of formula (11) is provided by Appendix B.3. Among others, formula (11) allows to approximate the asymptotic standard errors of the CLS estimators by plugging in the obtained estimates. Compared to the ML estimators of the previous Section 4.1, the CLS estimators have the advantage of being computed more easily (explicit formula instead of numerical maximization) and of having an explicit, data-independent expression for the asymptotic distribution, while the ML estimators usually have smaller standard errors within a class of estimating functions, see (Godambe & Heyde, 1987), as also indicated by the results of (Fokianos et al., 2009) concerning the INGARCH(1,1) model and by Table 5 below. A type of weighted CLS estimators, with the same asymptotic properties as the ML estimators of Section 4.1, was proposed by (Zhu& Wang, 2009), who also investigated its mean square error in a

  the sequel, we shall propose different types of confidence region based either on the CLS estimators (10), the asymptotic distribution of which is given by formula (11), or based on the ML estimator of Section 4.1, where the inverse of the observed Fisher information (7) is used to approximate the true covariance.2 We present results from a simulation study, where we investigated the performance of the confidence regions for time series of finite length. Explicit recommendations for the use of the confidence regions are derived.

  16) as the ML 2 confidence region. The shape of these regions can be recognized from Figures 5 (b) and (d), respectively. The practitioner is interested in the performance of these confidence regions for finite values of T , say T = 100, 200, 500 or 1000. To estimate the true coverage probability P ( I γ ∋ (β, α) ) of the confidence region I γ , where (β, α)

  Finally, we also compare the INARCH(1) model to different types of AR(1)-like model for time series of counts: the two-parametric equidispersed Poisson INAR(1) model and the three-parametric overdispersed negative binomial random coefficient thinning INAR(1) model (NB RCINAR(1)), negative binomial iterated thinning INAR(1) model (NB IINAR(1)) and generalized Poisson quasi-binomial thinning INAR(1) model (GP QINAR(

Page

  tion 4.4 for different values of γ; the results are shown in Figure 5. Obviously, the estimated value of α deviates significantly from 0 in all cases. The performance of these confidence regions can be recognized from the results of 29

Figure 4 :Figure 5 :

 45 Figure 4: Monthly strike data: (a) Line plot, (b) empirical autocorrelation and (c) partial autocorrelation function, (d) histogram with P o(4.94) fit.

  s n,j denote again the Stirling numbers of the first kind, see Proposition 2.2. In the case of the Poisson distribution P o(λ), one simply obtains k X (z) = λz, i. e., κ (1) = κ 1 = λ and κ (r) = 0 for r ≥ 2.

  Using κ (n) = k X (n) (0) and κ n = κ X (n) (0) completes the proof.Maximizing ℓ(β, α) is equivalent to maximizing l(β, α) = -(T -1)β -α ∑ T t=2 x t-1 + ∑ T t=2 x t • ln (β + αx t-1 ), (B.1)see formula (5). Taking partial derivatives, one obtains∂ ∂β ℓ(β, α) = -(T -1) + ∑ T t=2 x t /(β + αx t-1 ) x t x t-1 /(β + αx t-1 ) β • (a) + α • (b), it follows that β(T -1) + α ∑ T t=2 x t-1 = ∑ T t=2 x t (β + αx t-1 )/(β + αx t-1 ) = ∑ T t=2 x t . (B.3)Combining formulae (B.2) and (B.3), the proof of formula (6) is complete. Furthermore, formula (7) follows from formula (B.2) by taking again partial derivatives.

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	Page 18 of 59
	distribution:
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	function is easy to compute, since the process has a conditional Poisson
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Table 3 :

 3 Estimated coverage probabilities of the confidence region ML 1 : (14).

	T , as expected. But the
	22

Table 4 :

 4 Estimated coverage probabilities of the confidence region ML 2 : (16).

				γ = 0.90, T =		γ = 0.95, T =
	β	α	100	200	500 1000	100	200	500 1000
	2 0.2 0.921 0.921 0.923 0.926 0.958 0.960 0.960 0.962
		0.4 0.923 0.922 0.922 0.927 0.960 0.959 0.959 0.963
		0.6 0.920 0.926 0.927 0.925 0.959 0.960 0.963 0.962
		0.8 0.915 0.921 0.923 0.926 0.956 0.959 0.960 0.962
	4 0.2 0.930 0.929 0.931 0.930 0.964 0.963 0.964 0.965
		0.4 0.931 0.932 0.931 0.931 0.964 0.964 0.964 0.965
		0.6 0.929 0.929 0.930 0.930 0.963 0.964 0.965 0.963
		0.8 0.920 0.926 0.929 0.932 0.958 0.961 0.964 0.966
	6 0.2 0.934 0.936 0.934 0.934 0.966 0.967 0.965 0.966
		0.4 0.933 0.935 0.935 0.934 0.965 0.967 0.967 0.965
		0.6 0.932 0.934 0.934 0.936 0.964 0.967 0.967 0.968
		0.8 0.924 0.931 0.932 0.934 0.959 0.964 0.965 0.967
				γ = 0.975, T =		γ = 0.99, T =
	β	α	100	200	500 1000	100	200	500 1000
	2 0.2 0.977 0.979 0.980 0.980 0.990 0.991 0.992 0.992
		0.4 0.979 0.979 0.980 0.981 0.991 0.991 0.992 0.992
		0.6 0.978 0.980 0.980 0.981 0.990 0.991 0.991 0.992
		0.8 0.977 0.979 0.980 0.981 0.990 0.991 0.992 0.992
	4 0.2 0.982 0.981 0.982 0.982 0.992 0.993 0.993 0.993
		0.4 0.982 0.981 0.982 0.982 0.992 0.992 0.993 0.993
		0.6 0.980 0.982 0.982 0.981 0.992 0.993 0.993 0.992
		0.8 0.978 0.979 0.982 0.982 0.990 0.992 0.993 0.992
	6 0.2 0.983 0.984 0.982 0.983 0.993 0.994 0.993 0.993
		0.4 0.982 0.983 0.983 0.984 0.993 0.993 0.993 0.994
		0.6 0.981 0.983 0.982 0.984 0.993 0.993 0.993 0.993
		0.8 0.979 0.982 0.982 0.982 0.991 0.993 0.993 0.993

Data Example: Counts of Strikes

  In those situations, where the true coverage probability of the CLS 1 region is too small, i. e., if T is small and α

	5 Real-
	F o r
	P
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	large, the approximate and rectangular Bonferroni regions (15) or (16) can
	be used as a further alternative, at least if γ ≥ 0.95.
	27

2, or if the sample size satisfies T ≥ 500 (0.4 ≤ α ≤ 0.6) or even T ≥ 1000 (α ≈ 0.8); it is interesting to note that

(Fokianos et al., 2009)

found similar restrictions for the sample size concerning the INGARCH(1,1) model.

  Table 5, are close to each other; the numbers shown in parentheses are the respective asymptotic standard errors computed as described in Section 4.

	close to the empirical variance 7.92. Also an analysis of the corresponding
	empirical residuals
	εt
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	The variance of the ML estimated INARCH(1) model equals 8.37, being

Table 5 :

 5 Estimated parameters of several models.

	Model	Method Par. 1	Par. 2 Par. 3 AIC BIC
	INARCH(1)	MM 0.5735 2.1090
	(α, β)	(0.0881)	(0.4277)
		CLS 0.5841 2.0716
		(0.0876)	(0.4273)
		ML 0.6364 1.8114	464.3 469.7
		(0.0807)	(0.3860)
	Po INAR(1)	ML 0.5061 2.4603	473.1 478.5
	(α, λ)	(0.0561)	(0.2989)
	NB RCINAR(1)	ML 0.5918 10.1779 0.6701 467.1 475.2
	(ρ, n, p)	(0.0632)	(4.8583)	(0.1076)
	NB IINAR(1)	ML 0.6308 7.7564 4.2181 466.8 474.8
	(ρ, n, α)	(0.0797)	(3.8682)	(1.5811)
	GP QINAR(1)	ML 0.5928 4.0912 0.1841 466.9 475.0
	(ρ, λ, θ)	(0.0627)	(0.4826)	(0.0667)

mcmaster.ca Communications in Statistics -Simulation and Computation

  Du, J.-G., Li, Y. (1991). The integer-valued autoregressive (INAR(p)) model. Jour.

	Page 34 of 59 Page 36 of 59
	6 Conclusion and Future Work models), but which is also true for different model types like the INAR(p)
	models of (Du & Li, 1991). Time Series Analysis 12(2):129-142.
	In this article, we recommended the INARCH(1) model as a simple and parsi-
	moniously parametrized model for time series of overdispersed counts. After Ferland, R., Latour, A., Oraichi, D. (2006). Integer-valued GARCH processes.
	o r having reviewed the definition and basic properties of the INARCH(1) model, we showed how to approximate the marginal process distribution with the F o r Jour. Time Series Analysis 27(6):923-942. F r o Fokianos, K., Rahbek, A., Tjøstheim, D. (2009). Poisson autoregression. Jour.
	P e e r Amer. Stat. Assoc. 10(4):1430-1439. help of the Poisson-Charlier expansion. It became clear that this approxima-P P tion always leads to an improvement compared to a Poisson approximation, e Freeland, R.K. (1998). Statistical analysis of discrete time series with applications e but can be applied really satisfactorily only for INARCH(1) processes with an at most moderate degree of autocorrelation; otherwise, the more com-e r to the analysis of workers compensation claims data. PhD thesis, University of e r British Columbia, Canada.
	R e plex Markov chain method is preferable. Then we considered the problem of parameter estimation and derived explicit expressions for the asymptotic R e R e Freeland, R.K., McCabe, B. (2005). Asymptotic properties of CLS estimators in
	v i e w distribution of the ML and CLS estimators. Furthermore, we constructed v i e the Poisson AR(1) model. Stat. Prob. Letters 73(2):147-153. v i Friedman, D.J. (1993). Some considerations in the use of quality control techniques e several simultaneous confidence regions for the two model parameters, the performance of which is influenced by the length of the available time se-w w in integrated circuit fabrication. Int. Stat. Review 61(1):97-107.
	O n l ries and the degree of autocorrelation. A real-data example showed that the INARCH(1) model has a great potential for applications in practice. O n l Godambe, V.P., Heyde, C.C. (1987). Quasi-likelihood and optimal estimation. Int. O l n Stat. Review 55(3):231-244.
	y A particularly important issue for future research is to investigate if the y y Heimann, P.A. (1996). Attributes control charts with large sample sizes. Jour.
	Poisson-Charlier expansion might be applicable also to different types of Qual. Tech. 28(4):451-459.
	count data time series. For none of the INGARCH models, the marginal Heinen, A. (2003). Modelling Time Series Count Data: An Autoregressive Condi-
	distribution is explicitly known (comparable to the case of the usual GARCH
	32 34

URL: http://mc.manuscriptcentral.com/lssp E-mail: comstat@univmail.cis.URL: http://mc.manuscriptcentral.com/lssp E-mail: comstat@univmail.cis.mcmaster.

ca Communications in Statistics -Simulation and Computation

  Weiß, C.H. (2009). Modelling time series of counts with overdispersion. Stat. Meth.

	Appl. 18(4):507-519.
	Zhu, F., Wang, D. (2009). Estimation and testing for a Poisson autoregressive
	F o r model. To appear in Metrika. F Zhu, F., Wang, D. (2010). Diagnostic checking integer-valued ARCH(p) models o r using conditional residual autocorrelations. Comp. Stat. Data Analysis 54:496-
	508.	P P
		e e
		e r e r
		R R
		e e
		v i e v i e
		w w
		O n l O n l
		y y

Table A

 A

	.1.
	It becomes clear that the first order approximation is just the Poisson approxima-

tion, i. e., the true probabilities p i are approximated by the Poisson probabilities π i 37 Page 39 of 59 URL: http://mc.manuscriptcentral.com/lssp E-mail: comstat@univmail.cis.mcmaster.

ca Communications in Statistics -Simulation and Computation

  Table A.1: The first four Poisson-Charlier approximations.

	F o r
	P
	e
	e r
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	v i e
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	Page 41 of 59
	see Proposition 2.2 and formula [1,5] on page 452 in
	F o r
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	1+α)(1+α+α 2 ) + 3β 1+α + β 2 )	,
	(B.4)
	39

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ca Communications in Statistics -Simulation and Computation

  Furthermore, also the following expectations exist, see (B.4):

																		Page 42 of 59
	W :=	   	E E	[( ∂ ∂β g t-1 [( ∂ ∂α g t-1	) )	• •	( ∂ ∂β g t-1 ( ∂ ∂β g t-1	) )	• u 2 t • u 2 t	] ]	E E	[( ∂ ∂β g t-1 [( ∂ ∂α g t-1	) )	• •	( ∂ ∂α g t-1 ( ∂ ∂α g t-1	)	• u 2 t	]
	F o r
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												O n l	7)
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Note that the CLS and ML estimates can be computed using the available R functions lm(data ) and glm(data, family=poisson(link="identity")), respectively.

Communications in Statistics -Simulation and Computation

Acknowledgements. The author would like to thank the three referees for many useful comments on an earlier draft of this article.


Francisco B Pereira

Pedro M R Guimarães

José A Teixeira

Lucília Domingues
email: luciliad@deb.uminho.pt

Selection of Saccharomyces cerevisiae strains for efficient very high gravity bio-ethanol fermentation processes

Keywords: Bio-ethanol production, Industrial strains, Laboratory strains, Saccharomyces cerevisiae, Very high gravity fermentation

An optimized very high gravity (VHG) glucose medium supplemented with low cost nutrient sources was used to evaluate bio-ethanol production by 11 Saccharomyces cerevisiae strains. The industrial strains PE-2 and CA1185 exhibited the best overall fermentation performance, producing an ethanol titre of 19.2 % (v/v) corresponding to a batch productivity of 2.5 g l -1 h -1 , while the best laboratory strain (CEN.PK 113-7D) produced 17.5 % (v/v) ethanol with a productivity of 1.7 g l -1 h -1 . The results presented here emphasize the biodiversity found within S. cerevisiae species and that naturally adapted strains, such as PE-2 and CA1185, are likely to play a key role in facilitating the transition from laboratory technological breakthroughs to industrial-scale bioethanol fermentations.

Introduction

The application of Very High Gravity (VHG) fermentation technology, i.e. the use of highly concentrated sugar substrates, for the industrial production of bio-ethanol has a number of benefits, including: decreased process water requirements and energy costs, increased overall plant productivity and higher ethanol concentrations in the fermentation product that allow considerable savings in energy for distillation [START_REF] Wang | Optimization of an ethanol production medium in very high gravity fermentation[END_REF]. Although many microorganisms have been exploited for ethanol production, Saccharomyces cerevisiae strains are the most widely used in large-scale industrial bioethanol fermentation, namely due to their ability to ferment a wide range of sugars and develop appropriate mechanisms to deal with the harmful stress conditions found in VHG processes [START_REF] Bai | Ethanol fermentation technologies from sugar and starch feedstocks[END_REF]. Nevertheless, within S. cerevisiae species a wide variety of characteristics is observed, in particular between the so-called industrial and laboratory strains.

The fermentation processes under harsh industrial environmental conditions, especially using VHG media, require industrial strains [START_REF] Silva-Filho | Yeast population dynamics of industrial fuel-ethanol fermentation process assessed by PCR-fingerprinting[END_REF]. These strains rapidly respond to the stress conditions by adjusting their metabolic activities and become adapted over a long time in their specific environment [START_REF] Zhao | Yeast flocculation: New story in fuel ethanol production[END_REF].

Genetically, these are usually polyploid, aneuploid, or even alloploid and genetic manipulations, although feasible, are rather complex and seldom reported in literature [START_REF] Hansen | Modification of biochemical pathways in industrial yeasts[END_REF]. In contrast, the laboratory strains are often haploid or diploid, well documented, and easily manipulated by using modern molecular biology and metabolic engineering tools [START_REF] Albers | A comparison of stress tolerance in YPD and industrial lignocellulose-based medium among industrial and laboratory yeast strains[END_REF]. Although attractive in this respect, the performance of laboratory strains is generally weak under the very harsh conditions found in VHG industrial fermentations.

Spontaneous fermentation of high sugar juice is the main step in traditional "cachaça" (typical Brazilian distilled beverage) production and that is usually carried out by a mixed culture of yeasts, with predominance of S. cerevisiae strains physiologically adapted to the harsh environmental conditions [START_REF] Schwan | Microbiology and physiology of Cachaca (Aguardente) fermentations[END_REF]. Some strains of S. cerevisiae isolated from Brazilian sugarcane-to-ethanol distilleries showed high tolerance to ethanol and combined high fermentation efficiency with prolonged persistence in the fermentation system. In recent years, such strains have been widely adopted by the industry [START_REF] Basso | Yeast selection for fuel ethanol production in Brazil[END_REF]) and one of the most successful examples is the PE-2 wild isolate currently used by ~ 30 % of Brazilian distilleries, generating ~ 10 % of the world's bio-ethanol supply [START_REF] Argueso | Genome structure of a Saccharomyces cerevisiae strain widely used in bioethanol production[END_REF].

The aim of this work was to evaluate and compare the performance of laboratory and industrial strains under VHG batch fermentation conditions, in order to select the strain(s) that allow obtaining the highest ethanol titre and productivity.

Materials and Methods

Yeasts

The S. cerevisiae strains used included eight industrial strains: three isolated from Brazilian bio-ethanol production plants -PE-2, CAT-1, VR-1 [START_REF] Basso | Yeast selection for fuel ethanol production in Brazil[END_REF])and five belonging to the UFLA collection (Federal University of Lavras, Brazil) isolated from "cachaça" fermentation processes -CA11, CA1162, CA1185, CA1187, CA155 (Table 1). The S. cerevisiae laboratory strains (routinely used in yeast research laboratories) included haploid S288C, haploid CEN.PK 113-7D and diploid CEN.PK 122 (Table 1). Stock cultures were maintained on YPD [1 % (w/v) yeast extract, 2 % (w/v) bacto-peptone and 2 % (w/v) glucose] agar plates at 4 ºC.

Media and Ferme ntations

The fermentation tests were performed in a previously optimized VHG medium [START_REF] Pereira | Optimization of lowcost medium for very high gravity ethanol fermentations by Saccharomyces cerevisiae using statistical experimental designs[END_REF]) consisting of 280-350 g glucose l -1 , 44.3 g corn steep liquor (CSL) l -1 , 2.3 g urea l -1 , 3.8 g MgSO 4 •7H 2 O l -1 and 0.03 g CuSO 4 •5H 2 O l -1 . Glucose syrup and CSL were kindly provided by a starch manufacturer (COPAM, Portugal) and autoclaved separately (121 ºC, 20 min). After autoclaving, the whole CSL was allowed to settle for 1 -2 days at 4 ºC and then centrifuged (15 min at 13100 x g) to remove the insolubles.

The main composition of the CSL used has been previously reported [START_REF] Pereira | Optimization of lowcost medium for very high gravity ethanol fermentations by Saccharomyces cerevisiae using statistical experimental designs[END_REF]. The nutrients concentrated stock solutions were sterilized by filtration and then added aseptically. Before inoculating the fermentation flasks the medium was aerated by stirring with a magnet (>850 rpm during 20 min), reaching >95% of air saturation.

The yeast for inoculation was grown in Erlenmeyer flasks filled with medium containing 50 g glucose l -1 , 20 g peptone l -1 and 10 g yeast extract l -1 . After incubation at 30 ºC and 150 rpm for 18 -22 h (OD 600 of 7 -7.5), the cell suspension was aseptically collected by centrifugation (10 min at 7500 x g, 4 ºC) and ressuspended in 0.9 % (w/v) NaCl to a concentration of 200 mg fresh yeast ml -1 . Then, the yeast cells were pitched at about 1.3x10 8 cells ml -1 into 40 ml of culture medium to start the fermentation.

Fermentations were done in 100 ml Erlenmeyer flasks fitted with perforated rubber stoppers enclosing glycerol-locks (to permit CO 2 exhaustion while avoiding the entrance of air) and incubated at 30 ºC with 150 rpm orbital agitation. The initial pH was adjusted to 5.5 with NaOH. The final pH was > 3.9 in all fermentations. The fermentation evolution was monitored by mass loss and samples for analyses were taken just at the ending point.

Analytical Procedures

Cell dry weight was determined using 20 ml samples of the yeast culture collected by centrifugation (10 min at 7500 x g, 4 ºC) in a pre-weighed dried tube and then washed with 20 ml of distilled water. The tube was dried overnight at 105 ºC and weighed again. Glucose, ethanol and glycerol were analyzed by HPLC, using a Varian MetaCarb 87H column eluted at 60 ºC with 0.005 M H 2 SO 4 at a flow rate of 0.7 ml min -1 , and a refractive-index detector.

Determination of ferme ntation parameters

Ethanol conversion yield was calculated by the ratio between the maximum ethanol concentration produced and the glucose consumed (difference between the initial and residual glucose concentrations). It was expressed as a percentage (%) of the theoretical conversion yield, i.e. the yield considering a production of 0.511 g of ethanol per g of glucose. Ethanol productivity was defined as the ratio between final ethanol concentration and total fermentation time (fermentation was considered to be complete when the weight of the flasks stabilized).

Results and Discussion

An optimized VHG medium (280-350 g glucose l -1 ) supplemented with low-cost nutrients was used to evaluate the fermentation performance of 3 standard laboratory strains and 8 industrial strains, 3 isolated from bio-ethanol distilleries and 5 from "cachaça" fermentations in Brazil.

Figure 1 shows the CO 2 production (mass loss) profiles for the 11 strains tested. These profiles permit the evaluation of fermentation evolution since under the oxygen-limiting conditions used in this study, the patterns of CO 2 and ethanol production are closely related, because most CO 2 originates from the fermentative pathway. The fermenta tion parameters determined at the end of the fermentations are shown in Figure 2.

Among the laboratory strains, CEN.PK 113-7D presented the best fermentation performance, producing 17.5 % (v/v) ethanol in less than 80 h. The behavior of the diploid version CEN.PK 122 was similar, with fermentation being slightly slower (Figure 1A). Fermentation by strain S288C was much slower, taking over 120 h to reach completion, but the final ethanol titre reached was only slightly lower (Figure 2A). The 3 strains were able to completely consume initial glucose concentrations close to 300 g l -1 (Figure 2C), with ethanol yields over 90 % of the theoretical (Figure 2B). However, when the initial glucose concentration was raised to about 330 g l -1 , fermentation by strain CEN.PK 113-7D was slower and stopped with 36 g of residual glucose l -1 (data not shown). Hence, this increase in the initial glucose concentration resulted in a decrease in the fermentation rate and, consequently, in the overall ethanol productivity (which dropped to 1.5 g l -1 h -1 ), probably due to the increased osmotic pressure (higher content of sugar) in the beginning of the fermentation.

The industrial strains were tested with higher glucose concentrations (ca. 330 g l -1 ) (Figure 2C). Among the strains isolated from bio-ethanol distilleries, strains PE-2 and VR-1 showed identical CO 2 production profiles, with fermentation reaching completion in less than 80 h, while strain CAT-1 was slower (Figure 1B). The 3 strains consumed nearly all glucose (residual < 1 g l -1 ) (Figure 2C) but PE-2 produced a higher final ethanol titre (19 % v/v) (Figure 2A) and consequently presented a better ethanol yield (90 % of theoretical) (Figure 2B). Among the strains isolated from "cachaça" environments, CA1185 and CA1187 showed similar CO 2 production profiles, with fermentation reaching completion in less than 80 h (Figure 1C), in agreement with the results obtained for strain PE-2 and VR-1 (Figure 1B). Fermentation by strain CA155 was slower but reached identical final CO 2 production, with total glucose consumption. These 3 strains produced over 18 % (v/v) ethanol (Figure 2A), with conversion yields over 85 % of theoretical (Figure 2B).

Fermentation by strain CA1162 was slower (Figure 1C), producing 17 % (v/v) ethanol but leaving a glucose residual of 13 g l -1 (Figure 2). Strain CA11 showed the slowest fermentation amongst the industrial strains, with fermentation being incomplete even after 140 h (Figure 1C; Figure 2C).

These results indicate that there is heterogeneity among these natural yeast isolates in their ability to ferment VHG glucose medium and in their response to stress conditions. Accordingly, using different molecular techniques [START_REF] Pataro | Yeast communities and genetic polymorphism of Saccharomyces cerevisiae strains associated with artisanal fermentation in Brazil[END_REF] observed high molecular diversity of the prevalent S. cerevisiae strains in "cachaça" production, which was attributed to a succession of indigenous strains that occur throughout the "cachaça" fermentation season and among different seasons.

In order to investigate the maximum ethanol titres that could be produced, strains PE-2, CA1185 and CA1187 were tested in fermentations with 350 g glucose l -1 (data not shown). These strains fermented nearly all the glucose (residual < 4 g l -1 ), reaching ethanol titres of 19.2 ± 0.3, 19.2 ± 0.0 and 18.1 ± 0.0 % (v/v), respectively. Strains PE-2 and CA1185 showed the higher ethanol productivity (2.5 g l -1 h -1 ), which was lower in the case of CA1187 (1.8 g l -1 h -1 ).

Laboratory strains showed a final biomass concentration around 7-8 g l -1 , which was lower than that obtained with industrial strains (8-11 g l -1 ) (Figure 2D). This higher growth of industrial strains on VHG conditions can be related to its origin, since these strains were isolated from stressful environments and were physiologically adapted to media with high osmotic pressure and ethanol stress [START_REF] Basso | Yeast selection for fuel ethanol production in Brazil[END_REF].

There were no considerable differences in glycerol biosynthesis by the laboratory strains with a concentration around 8 g l -1 being obtained at the end of the fermentations, corresponding to 5-6 % of the final ethanol concentration (Figure 2D).

Interestingly, the industrial strains produced proportionately more glycerol, which accumulated to 8-10 % of the final ethanol concentration (Figure 2D). This increase in glycerol levels may contribute to counteract the effect of increasing osmotic pressure, promoting a protective effect on yeast cells [START_REF] Kaino | Gene expression profiles and intracellular contents of stress protectants in Saccharomyces cerevisiae under ethanol and sorbitol stresses[END_REF].

Increasing ethanol concentration at the end of fermentation (by applying high content of substrate) can significantly save the energy consumption in the downstream processes [START_REF] Zhao | Yeast flocculation: New story in fuel ethanol production[END_REF]. Nevertheless, the osmotic pressure and ethanol inhibition developed under VHG conditions frequently result in incomplete or stuck fermentations with consequent losses in productivity and raw material. However, our results showed that robust strains isolated from stressful industrial environments, such as PE-2 and CA1185, were able to consume over 330 g glucose l -1 and produce high ethanol titres (up to 19 %) with high ethanol batch productivity (> 2.3 g l -1 h -1 ). The results suggest that these strains are more prepared to cope with VHG fermentation stresses, namely high ethanol concentrations and osmotic pressure, comparing to standard laboratory yeast strains. These industrial yeasts must sense and rapidly adapt to the adverse factors found in their natural environment, adjusting their physiology and metabolic activities to avoid substantial viability loss in the culture. [START_REF] Silva-Filho | Yeast population dynamics of industrial fuel-ethanol fermentation process assessed by PCR-fingerprinting[END_REF] concluded by PCRfingerprinting of yeast samples from fuel ethanol fermentations that indigenous strains could be more adapted to the industrial process than commercial ones. Probably, during yeast cell recycling, selective pressure (an adaptive evolution) is imposed on yeast population, leading to strains with higher tolerance to the stressful conditions of the industrial fermentation. The results of the present study show that the great biodiversity found in distillery environments can be an important source of robust stress-tolerant strains for biotechnological fermentation processes. These robust strains also constitute interesting hosts to be used in improvement programs through metabolic engineering.

Ethanol concentrations above 18 % (v/v), as obtained with strains PE-2, CA1185, CA1187 and CA155 (Figure 2A), are rarely reported in the literature. Additionally, the ethanol productivities attained (> 2.3 g l -1 h -1 ) are very interesting for batch VHG fermentation. To our knowledge, the highest ethanol titres (20.6 -23.8 % v/v) have been reported in fermentations of VHG wheat mashes at temperatures not higher than 27 ºC (see Jones and Ingledew 1994 and references therein). In Brazil 70-80 % of the distilleries producing bio-ethanol from sugarcane employ a fed-batch process with high yeast cell concentrations and high fermentation temperature (above 30 ºC) achieving fermentation times of only 6-10 h and final ethanol concentrations up to 11 % (v/v) [START_REF] Brethauer | Review: continuous hydrolysis and fermentation for cellulosic ethanol production[END_REF]. Intensification of the fermentation process could lead to higher ethanol titres, which would be beneficial for process economics due to energy savings in the distillation process. In fact, in the United States, some corn-to-ethanol distilleries are already obtaining fermentation products with 16-18 % (v/v) ethanol [START_REF] Shapouri | USDA´s 2002 ethanol cost-of-production survey[END_REF].

Conclusions

The application of VHG fermentation for fuel ethanol production can improve the final ethanol concentration and save significantly energy consumption. Our results highlight industrial distillery environments as a remarkable source of robust yeast strains for biotechnological fermentation processes. The industrial strains PE-2 and CA1185 exhibited the best overall fermentation performance, efficiently fermenting 330-350 g glucose l -1 producing over 19 % (v/v) ethanol with a batch productivity up to 2.5 g l -1 h - 1 . Among the laboratory strains, the haploid CEN.PK 113-7D presented the best VHG fermentation efficiency, consuming 300 g glucose l -1 with a production of 17.5 % (v/v) ethanol, corresponding to a productivity of 1.7 g l -1 h -1 . These results are of practical importance for the selection of suitable yeast strains for the development of highly efficient industrial VHG bio-ethanol fermentation systems.

Figure 1 -Figure 2 -

 12 Figure 1 -Profiles of CO 2 production obtained in VHG fermentations: (A) ca. 300 g glucose l -1 by laboratory strains; (B) ca. 330 g glucose l -1 by industrial strains isolated from bio-ethanol distilleries; (C) ca. 330 g glucose l -1 by industrial yeast strains isolated from

Table 1 -

 1 S. cerevisiae strains used in this study.

	Strain	Available from	Reference/Source
	CEN.PK 113-7D	EUROSCARF a collection	van Dijken et al. 2000
	CEN.PK 122	EUROSCARF a collection	van Dijken et al. 2000
	S288C	ATCC b collection (n.º 26108)	Mortimer and Johnston 1986
	PE-2	LNF c	Basso et al. 2008
	VR-1	LNF c	Basso et al. 2008
	CAT-1	LNF c	Basso et al. 2008
	CA11	LNF c	R. Schwan
	CA1162	UFLA d collection	R. Schwan
	CA1185	UFLA d collection	R. Schwan
	CA1187	UFLA d collection	R. Schwan
	CA155	UFLA d collection	R. Schwan

a European Saccharomyces cerevisiae Archive for Functional Analysis (euroscarf@em.uni-frankfurt.de) b American Type Culture Co llect ion (www.atcc.org) c LNF Lat ino A mericana, Brazil (lnf@lnf.co m.br) d Federal Un iversity of Lavras, Brazil (rschwan@dbi.ufla.br)

Acknowledgements

The authors thank COPAM -Companhia Portuguesa de Amidos, S.A. (Portugal) for kindly providing glucose syrup and CSL, and Whasley Duarte and Rosane Schwan (Federal University of Lavras, Brazil) for kindly providing the yeast strains (isolated from Brazilian "cachaça" and bio-ethanol fermentations). The financial support of Fundação para a Ciência e a Tecnologia (FCT), Portugal, is acknowledged: project ProBioethanol PTDC/BIO/66151/2006, grant SFRH/BD/64776/2009 to F. B. Pereira and grant SFRH/BPD/44328/2008 to P. M. R. Guimarães.


G L Riley

C R Thomas
email: c.r.thomas@bham.ac.uk

Applicability of Penicillium chrysogenum rheol ogical correlations to broths of other fungal strains

Keywords: Consistency index, Fungi, Morphology, Rheology C.R. Tho mas

Existing correlations of Power law consistency index with Penicillium chrysogenum biomass concentration and morphology were revised using a microscope magnificat ion of 50 times to characterize the latter, rather than the 80 times used previously. This allowed tests of the correlations on broths of Aspergillus oryzae and Aspergillus niger, which have such large mycelial sizes that a lower magn ification is required for accurate morphological analysis. The new correlations were successful at predicting the rheology of A. oryzae broths but not A. niger broths, which may be because of a change in the exponent on the biomass concentration in the correlations for the latter. Because the mean maximu m dimension of clu mps is magnification independent, the preferred correlat ion was:

where K is the consistency index (Pa s n ), C m is the biomass concentration as dry cell weight (gL -1 ) and D is the mean maximu m dimension ( m).

Introduction

The morphology of filamentous micro-organis ms in submerged fermentations plays a crucial role in determining their productivity [START_REF] Papagianni | Fungal mo rphology and metabolite production in submerged mycelial processes[END_REF]). The dispersed morphological form [START_REF] Paul | Characterisation of mycelial morphology using image analysis[END_REF] is common in industrial filamentous fermentations, especially in the production of antibiotics such as penicillin.

In such fermentations hyphal entanglement [START_REF] Metz | The rheology of mould suspensions[END_REF]) can result in significant changes in broth rheology i.e. the relationship between stress es and strains in the suspension. So metimes large increases in broth (apparent) viscosity are observed and broths often become "pseudoplastic" or "shear thinning" i.e. the apparent viscosity decreases as the shear rate increases . The results of high viscosity and pseudoplastic behaviour can be a reduction in fermenter mixing perfo rmance and process control (A manullah et al. 2004), with a consequential decrease in productivity and/or the production of undesirable metabolites [START_REF] Papagianni | Fungal mo rphology and metabolite production in submerged mycelial processes[END_REF]).

The interactions between morphology, broth rheology and fermenter performance are co mplex [START_REF] Metz | The rheology of mould suspensions[END_REF] but are amenable to study because morphology can be characterised by image analysis [START_REF] Paul | Characterisation of mycelial morphology using image analysis[END_REF]. This method has allowed useful correlations to be sought between morphology and rheology.

Pseudoplasticity is usually assumed in such studies so that the shear stress τ is related to the shear rate γ by a Power law relationship:

n K (1)
where the rheological parameters K and n are called the consistency index and the flow behaviour index respectively.

In an extensive study on an industrial strain of Penicillium chrysogenum, [START_REF] Riley | Effect of bio mass concentration and mycelial morphology on fermentation broth rheology[END_REF] proposed an average value of the flow behaviour index o f 0.35 was proposed for broths of >10 g (dry cell weight)L -1 .

Successful predictions of the Power law consistency index were possible using correlations with the biomass concentration and the (dispersed form) morphological parameters mean mycelial pro jected area and mean maximu m d imension (Pau l and Thomas 1998). The correlations were:

K = C m 2 × (1×10 -6 A + 3 × 10 -3 ) (2) K = C m 2 × (5×10 -5 D -10 -3 ) (3)
where K is the consistency index (Pa s n ), C m is the biomass concentration as dry cell weight (gL -1 ), D is the mean maximu m dimension ( m) of the mycelia and A is their mean projected area ( m 2 ). The morphology was characterised by image analysis according to the method described in [START_REF] Paul | Characterisation of mycelial morphology using image analysis[END_REF].

For the design and control of submerged fermentations of fungi growing in dispersed form, it would be very useful to have correlations applicable to all or many strains, for example Aspergillus niger and Aspergillus oryzae as well as P. chrysogenum. Clu mps of the Aspergillus strains studied here were generally much larger than those of P. chrysogenum and consequently a lower magnification was required for image analysis, if truncation of the mycelia l clu mps by the image frame was to be avoided.

Magnification has a strong influence on the measured mean projected area in particular, so correlations

(2) and (3) were revised for a lo wer magnification and then tested on A. niger and A. oryzae broths.

Materials and methods

Fermentations and morphological characterization

Penicillium chrysogenum P. chrysogenum fermentations and morphological analysis were carried out as detailed in [START_REF] Riley | Effect of bio mass concentration and mycelial morphology on fermentation broth rheology[END_REF] except a microscope magnification of 50 times was used.

Aspergillus oryzae

An industrial strain of A. oryzae (Novo Nordisk, Lyngby, Den mark) was grown on a semidefined mediu m in 5.8 l (AO1), 10.8 l (AO2) and 60 l (AO3) working volume batch fermentations.

Further details can be found in [START_REF] Riley | A study of the rheology of fungal fermentation broths[END_REF]. Characterization of mycelial mo rphology was by the method of [START_REF] Riley | Effect of bio mass concentration and mycelial morphology on fermentation broth rheology[END_REF] using a 50 t imes magnification. The concentration of mycelia on the slide was around 0.25 g l -1 after appropriate d ilution with a 20% (w/v) sucrose solution.

Aspergillus niger

Samples were taken fro m two 600 l A. niger (A 1233/citrate, IMI nu mber 075353) batch fermentations (AN1 and AN2) reported by [START_REF] John | A novel reactor with two independently driven impellers for gas liquid processing[END_REF]. The vessel contained an independently-driven, dual impeller system with draft tube. Only limited monosepsis could be achieved in the vessel but a pH of 2 reduced the contamination risk and ensured dispersed growth. Full details of these fermentations are given in [START_REF] John | A novel reactor with two independently driven impellers for gas liquid processing[END_REF]. Two 4 l working volu me aseptic batch fermentations were carried out at pH 2, 25 C, impeller diameter 100 mm, agitation rate 800 rp m (AN3) or 200-750 rp m (AN4), aeration rate 3 l min -1 . The mediu m was that of [START_REF] Roehr | Kinetics of bio mass format ion and citric acid production by Aspergillus niger on pilot plant scale[END_REF]. Morphology was characterized as for A. oryzae, with a magnification of x 50 and a biomass concentration on the slide of around 0.25 gl -1 after dilution with 20% (w/v) sucrose.

Morphological analysis was carried out on a Quantimet 570 ( Leica Cambridge Ltd., Camb ridge, UK) with a Polyvar optical microscope (Reichert Jung Opstiche Werke A G, Austria)

Rheology

The rheological characteristics of the fermentation broths were determined using a Rheomat 30 (Contraves, Zurich, Swit zerland) fitted with a six-b lade, 53 mm diam, Rushton turbine rotating in a 600 ml glass beaker containing 400 ml samp le [START_REF] Riley | Effect of bio mass concentration and mycelial morphology on fermentation broth rheology[END_REF]. The shear rate range was 3.3 s -1 to the maximu m possible value whilst the flow remained laminar. This depended on the sample, but never exceeded 75.8 s -1 . The sample temperature was controlled at 30 C 1 C for A. oryzae, and 23 1 C for P. chrysogenum and A. niger, i.e. close to the fermentation temperature.

Correlations

Penicillium chrysogenum

For rev ision of the P. chrysogenum correlat ions, the dependence of the consistency index K on the biomass concentration C m was not altered (i.e. K α C m 2 ) because this parameter should be magnificat ion independent. Using 50 times morphological data, linear regression was then us ed to find correlations between K/C m 2 and the mean projected area or the mean maximu m dimension. The new correlations were tested on Aspergillus broths.

Dependence of rheology on biomass concentration was also investigated to see if the exponent (α) of 2 on C m in the correlat ions applied to Aspergillus broths. Even small changes, even minor, in the exponent might result in failure of the correlations. Reconstitution experiments similar to those undertaken by [START_REF] Riley | Effect of bio mass concentration and mycelial morphology on fermentation broth rheology[END_REF] on P. chrysogenum broths were carried out on broth samples taken at the end of the rapid growth phase of fermentation AO1 and at various times fro m fermentations AO3, AN1 and AN2. These were reconstituted to biomass concentrations between 6-30 g L -1 , the rheology measured, and the exponent values found.

Results and discussion

Consistency index of Penicillium chrysogenum (50 t imes magnification)

Fig. 1 shows the morphological parameters mean mycelial projected area and mean maximu m dimension measured at 50 t imes magnification for the P. chrysogenum fermentations. The higher magnificat ion data of [START_REF] Riley | Effect of bio mass concentration and mycelial morphology on fermentation broth rheology[END_REF] are given fo r co mparison. The differences between the mean projected areas are clear, but differences between the mean maximu m dimensions were small. The lower areas at higher magnification are because more gaps between the hyphae are detected and existing gaps become larger.

Fig. 2 shows K/C m versus the mean projected area and the mean dimension at the lower magnificat ion. Table 1 shows the slopes and intercepts of the regression lines. The correlations for a magnificat ion of 50 times were:

K = C m 2 × [5 × 10 -7 A + 3 × 10 -3 ] (4) K = C m 2 × [4 × 10 -5 D -9 × 10 -4 ] ( 5
)
where the symbols have the same meanings as earlier. It can be seen that many of the data points in Fig. 2 are clustered between values of K/ C m of 0.4 and 0.6. As discussed in [START_REF] Riley | Effect of bio mass concentration and mycelial morphology on fermentation broth rheology[END_REF], these points were mainly measurements taken after the end of the rapid growth phase, when most of the samples were taken due to the extended nature of fed-batch fermentations. Measurements made early in the fermentations when mean projected area and maximu m dimension were high have a strong influence on the form of the regression lines. [START_REF] Riley | Effect of bio mass concentration and mycelial morphology on fermentation broth rheology[END_REF] d iscussed this and considered the possibility of having separate correlations for low and high mycelial sizes. However, they decided that the form of correlations ( 4) and ( 5) would be of most practical use given the imprecision of mo rphological data. Th is approach was also accepted here.

As the mean maximu m dimension D is nearly independent of magnificat ion (Fig. 1), it was expected that there would be no significant difference in the relationship between K/C m 2 and the mean maximu m dimension whatever the magnificat ion. Correlat ions (3) and ( 5) g ive similar p redictions within the range of their validity. That this was so can be seen fro m the values of the slopes and intercepts in Table 1, with no differences found at the 95% confidence level. A larger and significant difference was observed in the relationships between K/C m 2 and the mean projected areas (correlations ( 2) and ( 3)), as this parameter is magnificat ion dependent. The intercepts of the regression lines were identical whereas the slopes differed. The most likely reason for this is that the mycelia towards the end of the fermentation were smaller and less complex and were therefore less influenced by magnification changes, leading to the same intercept. When the mycelia were larger and more co mplex, the higher magnification permitted discrimination of gaps between hyphae that were not apparent at lower magnifications, leading to apparently lower pro jected areas.

Tests of the consistency index correlat ions on A. oryzae and A. niger

Correlations (4) and ( 5) were tested on the A. oryzae and the A. niger fermentations. The predicted values of the consistency index were plotted against the measured values (Fig. 3 and4). The error bars on the predicted points represent the combined error on the biomass concentration, its exponent in the correlation, the mean projected area or mean maximu m dimension and the constants of the correlation. Error analysis is discussed more fully in [START_REF] Riley | Effect of bio mass concentration and mycelial morphology on fermentation broth rheology[END_REF]. A reasonable agreement was observed between predicted and measured values of the consistency index for A. oryzae broths, but neither correlat ion described the rheological behaviour of A. niger broths well.

Values for A. niger and A. oryzae

The mean values and their standard deviations, standard errors on the mean, and the number of reconstitution experiments fro m which these were determined are given in Tab le 2. The mean values were tested for significant differences at the 95 % level. There was no significant difference between the mean value for P. chrysogenum and that for A. oryzae, but a significant difference was found for A. niger. As is an exponent, even small changes would have a significant effect on predictions. It is therefore unsurprising that correlations (4) and ( 5) are poor for A. niger broths. However, it is not possible to conclude that the failure of the correlations for A. niger broths was solely due to a variation in , as a different morphological relat ionship may also exist. It may be that varies so much between batch and fed batch fermentations of this organism [START_REF] Olsvik | Rheology of filamentous fermentations[END_REF]) that t o develop useful rheological correlations would require extensive and specific studies of A. niger fermentations. These

were not undertaken in this work.

As mentioned earlier, it would be very useful to have rheological correlations applicable to all or many fungal strains, to be used in the design and control of their submerged fermentations. Because of unpredictable changes in , this is clearly not yet achievable even with respect to just one of the rheological parameters i.e. the consistency index.

Flow behaviour index

As for P. chrysogenum [START_REF] Riley | Effect of bio mass concentration and mycelial morphology on fermentation broth rheology[END_REF], no simp le relat ionship was found between the flow behaviour index and the bio mass concentration for A. oryzae or A. niger. For P. chrysogenum broths with a bio mass concentration above 10 g L -1 , [START_REF] Riley | Effect of bio mass concentration and mycelial morphology on fermentation broth rheology[END_REF] proposed that an average value of 0.35 0.1 be used for the flow behaviour index. The values of the flow behaviour index were much higher for the more viscous A. oryzae broths (around 0.5) and were generally lower fo r A. niger around 0.2 (Fig. 5). As the flow behaviour index is an exponent in the Power law, the inability to find any global value for this parameter is disturbing and probably means that separate flow behaviour and consistency index correlations will be need for every species, and possibly for d ifferent strain s.

It should be noted that [START_REF] Petersen | Mult ivariate models for pred iction of rheological characteristics of filamentous fermentation broth from the size d istribution[END_REF] have used a light scattering technique to determine biomass particle size distributions in Aspergillus oryzae broths from 550 L fermentations and to develop useful rheological correlations . This fast and practical technique does not classify particles into morphological forms but uses principal co mponent analysis to extract size info rmation fro m the data. A mean value of the flo w behaviour index of about 0.4 was observed, which d id not depend on the principal components. This is lower than the value reported here for unknown reasons. Long run, this light scattering technique may be superior to image analysis for rheological studies on fungal broths but it needs further testing on other strains and fermentation conditions.

Conclusions

Correlations ( 4) and ( 5) obtained by using lower magnification P. chrysogenum morphological data were successful in predict ing the consistency index of A. oryzae fermentation broths at the same magnificat ion but gave poor predictions for A. niger broths. It is promising that these correlations worked well for the A. oryzae broths, as these mycelia were much larger and well outside the morphological range used to develop the correlations. The failure for A. niger broths and the requirement for different flow behaviour index values for the different species means that a globally applicable method to predict fungal broth rheology remains elusive, despite the power of morphological characterization by image analysis.

Further progress will probably require a far better understanding of how mycelial interactions in broths lead to high viscosities and non-Newtonian behaviour.

Because mean maximu m d imension was not magnification dependent, no significant difference was found between the predictions of correlations (3) and ( 5). Because they are magnification independent, these correlations are preferred.

Table 1: Slopes, intercepts and standard deviations for the regression of K/C m against mean projected area A and mean maximu m d imension D (Fig. 2).

K/C m 2 versus: slope intercept mean pro jected area (50 times) 5×10 -7 ± 6 x 10 -8 3 x 10 -3 ± 4 x 10 -4 mean pro jected area (80 times) 10 -6 ± 10 -7 3 x 10 -3 ± 4 x 10 -4 mean maximu m dimension (50 t imes) 4×10 -5 ± 6 x 10 -6 -9 x 10 -4 ± 10 -3 mean maximu m dimension (80 t imes) 5×10 -5 ± 8 x 10 -5 -1 x 10 -3 ± 10 -3

Fig. 1 :Fig. 3 :Fig. 4 :Fig. 5 :

 1345 Fig. 1: Mean projected areas and maximu m dimensions for Penicillium chrysogenum fermentations F1

Table 2 :

 2 Su mmary of mean exponent α on biomass for Penicillium chrysogenum, Aspergillus niger and

	Aspergillus oryza., σ: standard deviation, σ m :
	standard error on the mean and N: number of
	reconstitution experiments.
		Penicillium	Aspergillus	Aspergillus
		chrysogenum	oryzae	niger
	mean	2.0	1.7	2.7
	σ	0.3	0.2	0.3
	σ m	0.1	0.1	0.1
	N	20	4	7

Acknowledgements

This work was funded by the Biotechnology and Biological Research Council, UK. Novo Nordisk Lyngby, Den mark is thanked for an industrial strain of Aspergillus oryzae, and Dr P.W Co x for morphological analysis of Aspergillus niger.