Fig. 2. Fits of the jointed Michaelis – Menten type JMM and inverse exponential polynomial IEP models columns to
data for fruit harvested from growers 2 – 4 rows during the main harvest in 1991.
The JMM and IEP models both characterised the softening behaviour of fruit well Figs. 1 – 3.
Of the two, the IEP was the superior as it de- scribed
the entire
softening behaviour
well whereas the JMM tended to underestimate firm-
ness during the latter stages of softening. The relative standard errors of the parameter estimates
of the JMM model were consistently greater than those of the IEP model Tables 1 and 2. In
particular, estimates of the first rate parameter in the JMM model v
1
had very large errors. Across growers, the estimates and errors of the
parameters in the IEP model were reasonably consistent, as were the correlations between them.
In contrast, the estimates and errors of the parameters in the JMM model varied more across
growers, especially the v
1
parameter indicating that this parameter is related to ‘at random’ dif-
ferences between batches caused by different growing conditions and harvest maturity.
Fig. 3. Fits of the jointed Michaelis – Menten type JMM and inverse exponential polynomial IEP models columns to
data for fruit harvested from growers 2 – 4 rows during the main harvest period in 1994.
3. Results
3
.
1
. Goodness of fit of models Generally, the CMM model poorly character-
ised the softening behaviour of kiwifruit Fig. 1. Curves generated by this model tended to decline
too sharply during the early stages of fruit soften- ing, tending towards asymptotes at firmness val-
ues significantly higher than the measured values. In contrast, the EXP model reasonably described
the early stages of softening but often it over-esti- mated firmness during the middle stages of soften-
ing, and under-estimated final firmness Fig. 1. The CG model accurately characterised softening
behaviour during the early and middle stages. However, the curves generated inherently have
strong asymptotes and did not describe decline in firmness prevalent during the latter stages of soft-
ening. On some occasions, the CG model also produced curves with too much initial curvature.
Table 1 Parameter estimates and their relative standard errors Rel. S.E. = standard error
parameter estimate obtained from fitting the jointed Michaelis–Menten type JMM and inverse exponential polynomial IEP models to the data for fruit harvested from each
of the four growers during the main harvest period in 1991 Parameter
Grower 1 Grower 2
Grower 3 Grower 4
Rel. S.E. Estimate
Rel. S.E. Estimate
Rel. S.E. Estimate
Rel. S.E. Estimate
JMM 0.093
53.1 a
0.185 44.21
47.46 0.143
51.6 0.071
0.496 281.24
v
1
3.983 45.98
92.62 1.536
29.69 0.511
0.072 35.05
0.154 34.21
19.71 0.121
v
2
25.56 0.067
0.097 27.64
u 0.295
30.64 25.97
0.237 25.32
0.093 IEP
0.019 102.35
d 0.055
90.21 84.41
0.050 88.08
0.031 0.067
− 2.025
b 0.162
− 2.880
− 2.140
0.096 −
2.320 0.089
0.060 0.0862
0.109 0.0886
0.1210 0.065
b
1
0.0950 0.066
b
2
− 0.00095
0.095 −
0.00065 0.161
− 0.00069
0.095 −
0.00070 0.101
Estimates have large 95 confidence intervals that include 0.
In nearly all cases, all the parameters within each of the JMM and IEP models were highly
correlated with each other Table 3. Correlations between parameters in the JMM model were
slightly more variable than those in the IEP model, especially the correlations of the parame-
ter v
2
with the parameters a and u.
3
.
2
. Estimation and prediction using the IEP model
Since the IEP model provided the best descrip- tion of trends in firmness in storage, relationships
between the estimates of various properties and deductions of the curves e.g. initial firmness and
Table 2 Parameter estimates and their relative standard errors RSE; standard errors
parameter estimates obtained from fitting the jointed Michaelis–Menten type JMM and inverse exponential polynomial IEP models to the data for fruit harvested from each of the
four growers during the main harvest period in 1994 Grower 2
Grower 3 Grower 4
Parameter Grower 1
Rel. S.E. Estimate
Rel. S.E. Estimate
Rel. S.E. Rel. S.E.
Estimate Estimate
JMM 0.148
54.02 0.117
50.485 0.132
48.956 a
0.166 44.02
34.399 72.89
1.126 38.17
1.348 1.303
66.64 1.141
v
1
0.118 v
2
34.43 30.78
0.146 0.126
35.51 0.106
38.26 28.28
0.194 1.904
38.47 30.15
0.183 u
33.638 0.179
IEP 0.053
d 81.646
85.69 0.089
0.073 93.57
0.061 79.19
− 1.979
0.208 −
2.226 0.171
b −
2.801 0.143
− 3.108
0.210 0.134
0.0796 0.169
0.0927 0.109
0.113 0.0860
0.0839 b
1
− 0.00059
0.155 −
0.00063 0.202
− 0.00060
b
2
− 0.00062
0.148 0.237
Estimates have large 95 confidence intervals that include 0.
Table 3 Correlations between parameter estimates obtained from fitting the jointed Michaelis–Menten type JMM and inverse exponential
polynomial IEP models to the data for fruit harvested from each of the four growers during the main harvest period in 1991 IEP
JMM v
1
v
2
u Parameter
d b
b
1
Parameter b
2
a Grower 1
d 1
a 1
1 b
− 0.93
0.84 v
1
1 0.78
1 b
1
− 0.70
− 0.94
v
2
1 −
0.55 0.90
0.56 1
b
2
− 0.97
0.61 u
0.85 −
0.97 1
Grower 2 d
1 1
a 1
b v
1
0.93 −
0.94 1
0.94 1
b
1
− 0.86
− 0.8
v
2
− 0.93
1 u
0.90 −
0.99 0.84
1 b
2
0.69 0.83
− 0.97
1 Grower 3
a d
1 1
1 b
− 0.95
0.92 v
1
1 −
0.84 v
2
0.93 1
b
1
− 0.78
− 0.93
1 0.91
0.81 1
b
2
u 0.66
− 0.99
0.81 −
0.96 1
Grower 4 1
a d
1 1
b 0.9
1 v
1
− 0.95
0.85 1
b
1
− 0.72
− 0.75
v
2
− 0.93
1 −
0.97 u
0.88 0.65
1 b
2
0.65 0.84
− 0.97
1
time to reach a given level of firmness generated by that model were examined with the purpose of
identifying any which might reasonably predict the storage life of fruit.
Firstly, we estimated the time taken for the flesh firmness of fruit to reach the industry export
threshold level of c. 10 N t
10 est
using parameter estimates obtained from fitting the model to each
full data set. In order to calculate the time taken for firmness to drop to a given value of x N t
x est
such as 10 N, Eq. 5 needs to be first written as a cubic function of time. For given values of
parameters, b , b
1
, b
2
and d, and a given level of firmness, FF
L
, the roots of the function provide a solution to t
x est
: t
x est
= − 3b
1
b
2 2
b −
ln b
FF
L
− 1
n
− b
1 3
b
2 3
n
1 3
− b
1
b
2
6 The time to reach 10 N ranged from 123 to 203
days. The t
10 est
values were then plotted against the corresponding estimates of initial firmness d
1 + expb ; Fig. 4. The two parameters were
poorly related, with a non-significant P = 0.625 linear regression, which accounted for only 3 of
the total sum of squares in t
10 est
. We also plotted
Fig. 4. Relationship between the estimated initial flesh firmness of fruit and the estimated time taken for the flesh firmness of
that fruit to reach the export threshold level for firmness of 10 N t
10 est
, as determined using the IEP model. Each point represents the average estimate for a grower batch. Outliers
have been omitted.
Fig. 5. Relationship between the estimated time taken for the flesh firmness of fruit to reach a given level t
x est
and the estimated time taken to reach 10 N t
10 est
, as determined using the IEP model. Each point represents the average estimate for
a grower batch. Two outliers have been omitted.
data set while rearrangement of Eq. 5 provided an estimate of b
as: b
. =
ln 5
n i = 1
FF
i
− FF
FF −
FF
i
exp b
1
t
i
+ b
2
t
i 2
+ b
2 2
t
i 3
3b
1 1n
7 Subsequently, the correlations between t
10 pred
and t
10 est
values Figs. 5 and 6 were found to be non-signifi- cant though they did increase in strength the less
data sets were curtailed i.e. from 30 – 90 days.
4. Discussion
