240
M. Gevrey . S. Lek' T.Oberdorff
"Pad method": The derivatives profile
Fig. 12.5 shows the results of the partial derivative method. that is three graphs
(one for each variable). which represent the partial derivative of the output with
respect to the input values versus the values of this input.
As it can be seen in Fig. 12.5a. the values of partial derivatives of ESR with
respect to TSR are nearly all positive or equal to zero for the whole range of TSR
values. The partial derivatives plot can be divided into three parts: (1) TSRs lower
than 50: the partial derivatives of ESR are equal to zero. there is no increase in
ESR for an increase in TSR; (2) TSRs between 50 and 150: the partial derivatives
values of ESR increase. An increase in TSR leads to an increase in ESR; (3) TSRs
higher than 150: the values of ESR partial derivatives stay positive but are lower.
An increase in TSR leads to a more moderate increase in ESR.
Conceming the plots of the ESR partial derivatives with respect to SAD and
NPP. Fig. 12.5b and Fig. 12.5c respectively. both the points are mainly elose to
zero. There is an independence of the ESR variable versus SAD and NPP.
Fig 12.6 shows the results of the partial derivative of TSR with respect to the
two variables SAD and NPP.
Fig. 12.6a is the graph of the TSR partial derivatives with respect to SAD
versus SAD. All the partial derivative values are positive. In the case of the low
SAD values the partial derivatives values are high. and become lower with the
increase in SAD.
Fig. 12.6b shows the graph of the TSR partial derivatives with respect to NPP
versus NPP. The partial derivatives plot can be divided into three parts: (1) NPPs
lower than 1500 kg- 2 y"\ the partial derivatives are near zero. there is an
independence of NPPs for these values; (2) NPPs between 1500 and 2250 kg-V 1 •
the partial derivatives are positive and high. an increase in NPP leads to an
increase in TSR; (3) NPPs higher than 2250 kg-V'. the partial derivatives of TSR
are negative. an increase in NPPs leads to a decrease in TSR.
The relative contributions
The algorithm permitted the calculation of a value of SSD per variable. Several
simulations were undertaken to obtain those SSDs. Because there was variation
between simulations in the obtained values of SSD and the relative proportions
between the SSDs of the 3 variables were conserved. thus the SSDs were
expressed as a percentage of the sum of the three SSDs and then averaged. The
values obtained are: SSD TsR =76.77%. SSD SAD =19.74% and SSD NPP =3.49%.
Therefore. TSR is the most significant variable. followed by SAD and then NPP.
wh ich has a very low contribution. The differences between the three SSDs are
sufficient for the t-test to be significant (P
NPP as the inputs. The results obtained are: SSD SAD =70.67% and SSD NPP =29.33%.
The t-test gives a significant difference between SAD and NPP (P
M. Gevrey . S. Lek' T.Oberdorff
"Pad method": The derivatives profile
Fig. 12.5 shows the results of the partial derivative method. that is three graphs
(one for each variable). which represent the partial derivative of the output with
respect to the input values versus the values of this input.
As it can be seen in Fig. 12.5a. the values of partial derivatives of ESR with
respect to TSR are nearly all positive or equal to zero for the whole range of TSR
values. The partial derivatives plot can be divided into three parts: (1) TSRs lower
than 50: the partial derivatives of ESR are equal to zero. there is no increase in
ESR for an increase in TSR; (2) TSRs between 50 and 150: the partial derivatives
values of ESR increase. An increase in TSR leads to an increase in ESR; (3) TSRs
higher than 150: the values of ESR partial derivatives stay positive but are lower.
An increase in TSR leads to a more moderate increase in ESR.
Conceming the plots of the ESR partial derivatives with respect to SAD and
NPP. Fig. 12.5b and Fig. 12.5c respectively. both the points are mainly elose to
zero. There is an independence of the ESR variable versus SAD and NPP.
Fig 12.6 shows the results of the partial derivative of TSR with respect to the
two variables SAD and NPP.
Fig. 12.6a is the graph of the TSR partial derivatives with respect to SAD
versus SAD. All the partial derivative values are positive. In the case of the low
SAD values the partial derivatives values are high. and become lower with the
increase in SAD.
Fig. 12.6b shows the graph of the TSR partial derivatives with respect to NPP
versus NPP. The partial derivatives plot can be divided into three parts: (1) NPPs
lower than 1500 kg- 2 y"\ the partial derivatives are near zero. there is an
independence of NPPs for these values; (2) NPPs between 1500 and 2250 kg-V 1 •
the partial derivatives are positive and high. an increase in NPP leads to an
increase in TSR; (3) NPPs higher than 2250 kg-V'. the partial derivatives of TSR
are negative. an increase in NPPs leads to a decrease in TSR.
The relative contributions
The algorithm permitted the calculation of a value of SSD per variable. Several
simulations were undertaken to obtain those SSDs. Because there was variation
between simulations in the obtained values of SSD and the relative proportions
between the SSDs of the 3 variables were conserved. thus the SSDs were
expressed as a percentage of the sum of the three SSDs and then averaged. The
values obtained are: SSD TsR =76.77%. SSD SAD =19.74% and SSD NPP =3.49%.
Therefore. TSR is the most significant variable. followed by SAD and then NPP.
wh ich has a very low contribution. The differences between the three SSDs are
sufficient for the t-test to be significant (P
NPP as the inputs. The results obtained are: SSD SAD =70.67% and SSD NPP =29.33%.
The t-test gives a significant difference between SAD and NPP (P