ISO/TS 18571:2024

Road vehicles - Objective rating metric for non-ambiguous signals

This document provides validation metrics and rating procedures to calculate the level of correlation between two non-ambiguous signals obtained from a physical test and a computational model and it is aimed at vehicle safety applications. The objective comparison of time-history signals of model and test is validated against various loading cases under different types of physical loads such as forces, moments and accelerations. However, other applications can be possible too, but are not within the scope of this document. NOTE Annex A gives some examples of the application of this document.

Véhicules routiers — Mesures pour l'évaluation objective de signaux non ambigus

General Information

• Status: Published
• Publication Date: 14-May-2024
• ICS: 43.040.01 - Road vehicle systems in general
• Technical Committee: ISO/TC 22/SC 36 - Safety and impact testing
• Drafting Committee: ISO/TC 22/SC 36 - Safety and impact testing
• Current Stage: 6060 - International Standard published
• Start Date: 15-May-2024
• Due Date: 12-Oct-2025
• Completion Date: 15-May-2024

Relations

• Revises: ISO/TS 18571:2014 - Road vehicles - Objective rating metric for non-ambiguous signals
• Effective Date: 15-Oct-2022

Overview

ISO/TS 18571:2024 - Road vehicles - Objective rating metric for non-ambiguous signals - defines a standardized, objective metric and rating procedures to quantify the level of correlation between time‑history signals from physical tests and computational models. Focused on vehicle safety applications, the Technical Specification supports model validation by comparing non‑ambiguous signals (e.g., forces, moments, accelerations) and provides validated procedures, score components and preprocessing guidance. The document also contains informative case studies (Annex A).

Key topics and technical requirements

• Purpose: Provide validation metrics and procedures for objective comparison of CAE and test signals in road vehicle safety.
• Metric components:
  • Corridor score - evaluates how the model signal stays within tolerance corridors around the reference signal.
  • Phase score - quantifies timing (time‑shift) differences between signals.
  • Magnitude score - measures amplitude agreement between signals.
  • Slope score - assesses agreement of derivative / rate‑of‑change behavior (new algorithm uses a nine‑point moving average to smooth slopes).
• Overall ISO rating: Combines the component scores into a single, interpretable rating.
• Pre‑processing requirements:
  • Synchronization of signals, appropriate sampling rate, and filtering (smoothing) prior to scoring.
  • Guidance on interval of evaluation and handling multiple responses.
• Algorithmic notes:
  • Revisions in the 2024 edition include expanded guidance on window size for Dynamic Time Warping (DTW) (recommended ~10% of data length) and improvements to slope scoring.
• Limitations: The specification documents signal types within scope and provides caveats on metric interpretation and validation boundaries.

Practical applications

• Model validation in CAE workflows for crash, impact and occupant safety simulations.
• Objective assessment of component or subsystem dynamic responses (forces, accelerations, moments).
• Verification of simulation fidelity during design iterations and regulatory or supplier exchanges.
• Use in correlation reports, design reviews, and to justify model predictive capability for safety decisions.

Who should use this standard

• CAE engineers and model validation specialists in the automotive industry
• Vehicle safety test laboratories and OEM validation teams
• Simulation analysts comparing test and model time‑history outputs
• Researchers and standards developers working on objective signal comparison

Related standards

• ISO/TR 16250 (contextual background on model validation methods)
• References and related guidance from CAE/validation communities (AIAA, ASME, DMSO) are noted in the introduction for broader model validation practices.

Keywords: ISO/TS 18571:2024, objective rating metric, non‑ambiguous signals, road vehicles, model validation, CAE, time‑history signals, corridor score, phase score, magnitude score, slope score, dynamic time warping.

Standards Content (Sample)

Technical
Specification
ISO/TS 18571
Second edition
Road vehicles — Objective rating
2024-05
metric for non-ambiguous signals
Véhicules routiers — Mesures pour l'évaluation objective de
signaux non ambigus
Reference number
© ISO 2024
All rights reserved. Unless otherwise specified, or required in the context of its implementation, no part of this publication may
be reproduced or utilized otherwise in any form or by any means, electronic or mechanical, including photocopying, or posting on
the internet or an intranet, without prior written permission. Permission can be requested from either ISO at the address below
or ISO’s member body in the country of the requester.
ISO copyright office
CP 401 • Ch. de Blandonnet 8
CH-1214 Vernier, Geneva
Phone: [phone]
Email: [e-mail]
Website: www.iso.org
Published in Switzerland
ii
Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
4 Abbreviated terms and symbols . 2
4.1 Abbreviated terms .2
4.2 General .2
4.3 Corridor score .2
4.4 Phase, magnitude and slope scores .3
4.4.1 General .3
4.4.2 Phase score .3
4.4.3 Magnitude score.3
4.4.4 Slope score .4
4.5 Overall ISO rating .4
5 General data requirements . 5
6 ISO metric . 5
6.1 General .5
6.2 Calculation of the overall ISO rating .5
6.3 Corridor score .6
6.3.1 General .6
6.3.2 Calculation .7
6.3.3 Step by step procedure .8
6.4 Phase, magnitude and slope scores .8
6.4.1 Phase score .9
6.4.2 Magnitude score.10
6.4.3 Slope score . 13
6.4.4 Step by step procedure . 15
7 Meaning of the overall ISO rating .15
8 Pre-processing of the data . .16
8.1 General .16
8.2 Synchronization of the signals .16
8.3 Sampling rate .16
8.4 Filtering .16
8.5 Interval of evaluation .17
9 Limitations .18
9.1 General .18
9.2 Type of signals .18
9.3 Metric validation . .18
9.4 Meaning of the results .19
9.5 Multiple responses .19
Annex A (informative) Case studies .20
Bibliography .92

iii
Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards
bodies (ISO member bodies). The work of preparing International Standards is normally carried out through
ISO technical committees. Each member body interested in a subject for which a technical committee
has been established has the right to be represented on that committee. International organizations,
governmental and non-governmental, in liaison with ISO, also take part in the work. ISO collaborates closely
with the International Electrotechnical Commission (IEC) on all matters of electrotechnical standardization.
The procedures used to develop this document and those intended for its further maintenance are described
in the ISO/IEC Directives, Part 1. In particular, the different approval criteria needed for the different types
of ISO document should be noted. This document was drafted in accordance with the editorial rules of the
ISO/IEC Directives, Part 2 (see www.iso.org/directives).
ISO draws attention to the possibility that the implementation of this document may involve the use of (a)
patent(s). ISO takes no position concerning the evidence, validity or applicability of any claimed patent
rights in respect thereof. As of the date of publication of this document, ISO had not received notice of (a)
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www.iso.org/patents. ISO shall not be held responsible for identifying any or all such patent rights.
Any trade name used in this document is information given for the convenience of users and does not
constitute an endorsement.
For an explanation of the voluntary nature of standards, the meaning of ISO specific terms and expressions
related to conformity assessment, as well as information about ISO's adherence to the World Trade
Organization (WTO) principles in the Technical Barriers to Trade (TBT), see www.iso.org/iso/foreword.html.
This document was prepared by Technical Committee ISO/TC 22, Road vehicles, Subcommittee SC 36, Safety
and impact testing.
This second edition cancels and replaces the first edition (ISO/TS 18571:2014), which has been technically
revised.
The main changes are as follows:
— more descriptions about window size for dynamic time warping were provided. Ten percent of data
length was used as window size;
— for slope score calculation, a modified algorithm was developed. In the new algorithm, a nine-point
moving average method was used to keep data point symmetry and slope curve smooth;
— in original Annex data sets, some time intervals were not consistent with variations at thousandth digit.
Now, data sets were cleaned up and time interval variations were eliminated. New rating results on
Annex data sets were provided and all figures and tables were updated accordingly.
Any feedback or questions on this document should be directed to the user’s national standards body. A
complete listing of these bodies can be found at www.iso.org/members.html.

iv
Introduction
Computer Aided Engineering (CAE) has become a vital tool for product development in the automobile
industry. Various computer programs and models have been developed to simulate dynamic systems. To
maximize the use of these models, the validity and predictive capabilities of these models are assessed
quantitatively. Model validation is the process of comparing CAE model outputs with test measurements in
order to assess the validity or predictive capabilities of the CAE model for its intended usage. The fundamental
concepts and terminology of model validation have been established mainly by standard committees
[2]
including the American Institute of Aeronautics and Astronautics (AIAA) , the American Society of
Mechanical Engineers (ASME) Standards Committees on verification and validation of Computational Solid
[3] [4]
Mechanics and Computational Fluid Dynamics and Heat Transfer , the Defense Modeling and Simulation
[5] [6]
Office (DMSO) of the U.S. Department of Defense (DoD) , the United States Department of Energy (DOE)
[19],[20]
and various other professional societies .
One of the critical tasks to achieve quantitative assessments of models is to develop a validation metric that
has the desirable metric properties to quantify the discrepancy between functional or time history responses
[7],[16],[17]
from both physical test and simulation result of a dynamic system . Developing quantitative model
[11],[12],[13],[15],[17],[18],[2
validation methods has attracted considerable researchers’ interest in recent years
3],[24],[26]
. However, the primary consideration in the selection of an effective metric should be based on the
application requirements. In general, the validation metric is a quantitative measurement of the degree of
agreement between the physical test and simulation results.
[1]
This document is the essential excerpt of the ISO/TR 16250 which provides standardized calculations of
the correlation between two signals of dynamic systems, and it is validated against multiple vehicle safety
case studies.
v
Technical Specification ISO/TS 18571:2024(en)
Road vehicles — Objective rating metric for non-
ambiguous signals
1 Scope
This document provides validation metrics and rating procedures to calculate the level of correlation
between two non-ambiguous signals obtained from a physical test and a computational model and it is
aimed at vehicle safety applications. The objective comparison of time-history signals of model and test is
validated against various loading cases under different types of physical loads such as forces, moments and
accelerations. However, other applications can be possible too, but are not within the scope of this document.
NOTE Annex A gives some examples of the application of this document.
2 Normative references
There are no normative references in this document.
3 Terms and definitions
For the purposes of this document, the following terms and definitions apply.
ISO and IEC maintain terminology databases for use in standardization at the following addresses:
— ISO Online browsing platform: available at https:// www .iso .org/ obp
— IEC Electropedia: available at https:// www .electropedia .org/
3.1
filtering
smoothing of signals by using standardized algorithms
3.2
level of correlation
similarity of two signals
3.3
interval of evaluation
time domain that is used to calculate the correlation between two signals
3.4
rating
calculated value that represents a certain level of correlation (3.2) (objective rating)
3.5
sampling rate
recording frequency of a signal
3.6
time sample
pair values (e.g. time and amplitude) of a recorded signal

3.7
time-history signal
physical value recorded in a time domain
Note 1 to entry: Time-history signals are non-ambiguous.
4 Abbreviated terms and symbols
4.1 Abbreviated terms
CAE Computer Aided Engineering
CORA CORrelation and Analysis
DTW Dynamic Time Warping
EEARTH Enhanced Error Assessment of Response Time Histories
SME Subject Matter Expert
4.2 General
C , Ct()
Analyzed signal (CAE signal)
T , Tt
()
Reference signal (test signal)
Δt
Interval between two-time samples
t
Time signal (axis of abscissa)
t
Time zero of an event (e.g. test, crash, impact)
t
Starting time of the interval of evaluation
start
t
Ending time of the interval of evaluation
end
N
Total number of sample points (e.g. time steps) between the starting time t and ending time
start
t
end
4.3 Corridor score
Z
Corridor score
Corridor score at time t (curve)
Zt()
k
Exponent factor for calculating the corridor score between the inner and outer corridors
Z
a
Relative half width of the inner corridor
b
Relative half width of the outer corridor
δ
Half width of the inner corridor
i
δ
Half width of the outer corridor
o
Lower/upper bounds of the inner corridor at time t (curve)
δ ()t
i
Lower/upper bounds of the outer corridor at time t (curve)
δ t
()
o
Ν
All natural numbers without zero
>0
Absolute maximum amplitude of the reference signal T
T
norm
4.4 Phase, magnitude and slope scores
4.4.1 General
ts ts
Truncated and shifted CAE curve
C , Ci()
ts+w ts
C Warped CAE curve of C
ts+d ts
C Derivative CAE curve of C
ts+d ts
C Derivative CAE curve of C after averaging
ts ts
Truncated and shifted test curve
T , Tj
()
ts+w ts
T Warped test curve of T
ts+d ts
T Derivative test curve of T
ts+d ts
T Derivative test curve of T after averaging
4.4.2 Phase score
E
Phase score
P
k Exponent factor for calculating the phase score E
P P
*
Maximum allowable percentage of time shift
ε
P
m
Time steps moved to evaluate the phase error
n Number of time shifts to get ρ
ε E
ρ Maximum cross correlation of all ρ m and ρ m
() ()
E L R
ρ m Cross correlation – signal is moved to the left
()
L
ρ ()m Cross correlation – signal is moved to the right
R
4.4.3 Magnitude score
E
Magnitude score
M
k Exponent factor for calculating the magnitude score E
M M
*
Maximum allowable magnitude error
ε
M
ε
Magnitude error
mag
n ts ts
Number of data samples of time shifted and truncated curves ( C and T )

d
Local cost matrix to perform the dynamic time warping
di(), j
Local cost function to perform the dynamic time warping
di, j
[] Cumulative cost matrix
tw
D
Dynamic time warping distance
TW
Di(), j
Cost of the optimal warping path
TWopt
i ts
Index number of time shifted and truncated CAE curve C
ts
i
l Index number of l -th warping path of curve C
j ts
Index number of time shifted and truncated test curve T
ts
j
l Index number of l -th warping path of curve T
k
Index number of any warping path

Number of data samples of the optimal warping path
k
w
Optimal warping path
w
The l -th warping path cell
l
4.4.4 Slope score
E
Slope (topology) score
S
k Exponent factor for calculating the slope score E
S S
*
Maximum allowable slope error
ε
S
ε
Slope error
slope
4.5 Overall ISO rating
R
Overall ISO rating
Weighting factor of the corridor score Z
w
Z
w Weighting factor of the phase score E
P P
w Weighting factor of the magnitude score E
M M
w Weighting factor of the slope score E
S S
r
Rank of the sliding scale of the ISO metric
Lower threshold of rank r
Sr ()
Clower
Upper threshold of rank r
Sr
()
Cupper
5 General data requirements
The metric described in this document requires non-ambiguous curves (e.g. time-history curves).
Furthermore, it is required that the reference curve Tt() and the evaluated curve Ct() are both defined
between starting time t and ending time t . Both curves shall have the same number of sample points
start end
N with a constant time interval Δt within the evaluation interval.
6 ISO metric
6.1 General
The approach of this document is to combine different types of algorithms to get reliable and robust
assessments of the correlation of two signals. The calculated score provides fair assessment for poor and for
good correlations of two signals. The two most promising metrics are identified in Reference [1], they are the
CORA corridor method and EEARTH. A combined metric based on the improved CORA corridor method and
EEARTH is then proposed for this document which has been fully validated using responses from multiple
vehicle passive safety applications.
Figure 6.1 shows the structure of the overall ISO metric. While the corridor method calculates the deviation
between curves with the help of automatically generated corridors, the EEARTH method analyses specific
curve characteristics such as phase shift, magnitude and shape. Hence, the ISO metric consists of the two
best available algorithms.
Figure 6.1 — ISO metric structure
6.2 Calculation of the overall ISO rating
The combination of the four metric ratings (corridor, phase, magnitude and slope) will provide a single
number R for the correlation of the analysed signals which represents the final overall objective rating. The
overall objective rating R is calculated by combining the separate sub-ratings of corridor ( Z ), phase ( E ),
P
magnitude ( E ) and slope ( E ). Four individual weighting factors are defining the influence of each metric
M S
on the overall rating [see Formulae (6.1) and (6.2)]. The corresponding weighting factors are shown in
Table 6.1.
Rw=⋅Zw+⋅Ew+⋅Ew+⋅E (6.1)
ZP PM MS S
w ++w w +=w 1 (6.2)
ZPMS
Table 6.1 — Weighting factors of the ISO sub-ratings
Parameter Value Description
w 0,4 Weighting factor of the corridor score
Z
w 0,2 Weighting factor of the phase score
P
w
0,2 Weighting factor of the magnitude score
M
w 0,2 Weighting factor of the slope score
S
6.3 Corridor score
6.3.1 General
The corridor metric calculates the deviation between two signals by means of corridor fitting. The two sets
of corridors, the inner and the outer corridors, are defined along the mean curve. If the evaluated curve C is
within the inner corridor bounds, a score of “1” is given and if it is outside the outer corridors bounds, the
score is set to “0”. The assessment declines from “1” to “0” between the bounds of inner and outer corridors
resulting in three different rating zones as shown in Figure 6.2. The compliance with the corridors is
calculated at each specific time t and the final corridor score Z of a signal is the average of all scores Zt()
at specific times t .
Key
1 rating = 0
2 0 < rating < 0
3 rating = 1
[9]
Figure 6.2 — Rating zones of the corridor metric (corridors of constant width)
[14]
The philosophy of the ISO approach is to use a narrow inner corridor and a wide outer corridor . It limits
the number of “1” ratings to only good correlations and gives the opportunity to distinguish between poor
and fair correlations. If the outer corridor is too narrow, too many curves of a fair or moderate correlation
would get the same poor rating of “0”, like signals of almost no correlation with the reference. Basically, the
width of the corridors can be adjusted in order to reflect the specific signal characteristic. The width can
be constant for the whole duration of the dynamic responses or vary at the different time intervals. This
document applies the most common approach of using constant corridor widths for the whole duration of
[1],[25]
the dynamic response .
6.3.2 Calculation
The parameters a and b define the relative half widths of the inner and the outer corridors. Both shall be
0 0
between “0” and “1”, and a shall be less than b . The absolute half widths of both corridors are defined as
0 0
the product of relative half width and the absolute maximum amplitude T of the reference signal T .
norm
Formula (6.3) shows the calculation of T and it is calculated within the interval of evaluation.
norm
TT=maxm{}in() ,(max T) (6.3)
norm
The absolute half width of the inner corridor (absolute distance from the reference signal to the outer
bounds of the inner corridor) is defined by Formula (6.4). The calculation of the absolute half width of the
outer corridors [see Formula (6.5)] is similar to that of the inner corridors.
δ =⋅aT 01≤≤a (6.4)
in00orm
δ =⋅bT 01≤≤baand on00orm 00
Based on these definitions the lower and upper bounds of the inner corridor are defined by Formula (6.6)
and the lower and upper bounds of the outer corridor are defined by Formula (6.7).
δδtT= t ± (6.6)
() ()
ii
δδ()tT= ()t ± (6.7)
oo
Formula (6.8) shows the calculation of the corridor score for the correlation between the reference signal T
and the analysed signal C at each evaluation time t . If the absolute difference between the signals T and C
is less than the half width of the inner corridor (δ ), then the score is set to “1”. The score is calculated by
i
Formula (6.8) when the absolute difference between both signals is in between δδ≤ Tt()−Ct() ≤ . If the
io
absolute difference between both signals is greater than the half width of the outer corridor (δ ), then the
o
score is set to “0”. The parameter k assesses the location of the analysed signal within the outer corridor,
Z
and it applies the appropriate penalty on the score. A linear ( k =1 ), quadratic ( k =2 ), cubical ( k =3 ) or
Z Z Z
any other regression relationship can be defined accordingly.
1 if Tt()−Ct() <δ

i

k
Z

δ −Tt()−Ct() 
o
Zt()= k ∈Ν (6.8)

 
Z >0
δδ−
 
oi


0 if Tt()−Ct() >δδ
o

The final corridor score Z is calculated by averaging all single time step score Zt() as shown in Formula (6.9).
The parameter N represents the total number of sample points (e.g. time steps) between starting and
ending times of the interval of evaluation.
t
end
Zt()
∑
tt=
start
Z= (6.9)
N
One of the advantages of the corridor metric is the simplicity and the clearness of the algorithm. It
reflects criteria which are used intuitively in engineering judgment. Sometimes this simplicity may be the
disadvantage of the method. For example, a small distortion of the phase can lead to a very undesirable
[1]
rating .
[14]
Based on a sensitivity study of CORA and as described in Reference [1], fixed width corridors are
employed and the most appropriate metric parameters are identified as shown in Table 6.2.

Table 6.2 — Parameters of the corridor metric
Parameter Value Description
a 0,05 Relative half width of the inner corridor
b 0,50 Relative half width of the outer corridor
k
2 Transition between ratings of “1” and “0” (progression)
Z
6.3.3 Step by step procedure
First, the signals shall be pre-processed as described in Clause 8. After preparing the signals for the analysis
and defining the interval of evaluation, the maximum absolute amplitude T of the reference signal T
norm
shall be determined within this interval. It is used to calculate the inner and outer corridors. The actual
corridor assessment shall be executed within this defined interval. The total score ranges between “0” and
“1”. A score of “1” does not mean that both signals are identical. Solely their correlation is mathematically
perfect within the defined tolerances.
To summarize, the following step-by-step procedures shall be followed to calculate corridor score:
a) Pre-process both signals according to Clause 8.
b) Calculate T within the interval of evaluation by using the reference signal.
norm
c) Calculate the inner and the outer corridors.
d) Calculate the corridor score Zt() at every specific time t within the interval of evaluation.
e) Calculate the total corridor score Z based on Zt() and the number N of time sample points.
6.4 Phase, magnitude and slope scores
Phase, magnitude and slope (or so-called topology) error assessments between the time history curves T
[23] [27]
and C are used as objective rating metrics in addition to the corridor metric described before. The
,
enhanced error assessment of response time histories (EEARTH) metric combines these three assessments
[27]
to the global response error . It is defined as the error associated with the complete time history with
equal weight on each point. Quantifying the errors associated with these features of phase, magnitude and
slope (topology) separately is challenging because there are strong interactions among them. For example,
to quantify the error associated with magnitude, the presence of a phase difference between the time
[22]
histories may result in a misleading measurement. A unique feature dynamic time warping (DTW) is
used to separate the interaction of phase, magnitude and slope (topology) errors. It aligns peaks and valleys
as much as possible by expanding and compressing the time axis according to a given cost (distance)
[9]
function .
The ranges of the three errors are quite different and there is no single rating that can provide a quantitative
assessment alone. Therefore, a numerical optimization method is employed to identify the appropriate
parameters so that the resulted phase, magnitude and slope sub-ratings can match with SME’s ratings
[8],[21]
closely . Figure 6.3 shows the workflow of the procedures and the details of the algorithms are
described in the following subsections.

Figure 6.3 — Workflow of the calculation of phase, magnitude and slope scores
6.4.1 Phase score
The phase score E is used to measure the phase lag between the two-time histories T and C . The maximum
P
*
allowable percentage of time shift is ε and it is pre-defined. In this step, the initial curve C is shifted left
P
then right one step at a time to the original test data, curve T , and the cross correlation between the
truncated test curve T , and shifted and truncated C are calculated until reaching the maximum allowable
*
time shift limits ε ⋅−()tt .
Pend start
When the initial curve C is moved to the left by m time steps, the number of overlapping points of the two
time histories after time shift mt⋅Δ is reduced to n (nN=−m ) and the corresponding cross correlation
value ρ ()m is calculated by Formula (6.10).
L
nn−1
[(Ct((++mi))⋅−ΔΔtC()tT)(⋅+()ti⋅−tT(t))]
∑ startstart
i=0
ρ ()m = (6.10)
L
n−1 n−1
2 2
[(Ct ++()mi ⋅−ΔΔtC)(tT)] ⋅+[(ti⋅−tT)(t))]
∑ starts∑ tart
i=0 i=0
When the initial curve C is moved to the right by m time steps, the number of overlapping points after time
shift mt⋅Δ is reduced to n (nN=−m ) and the corresponding cross correlation value ρ ()m is calculated
R
by Formula (6.11)
nn−1
[(Ct()+⋅itΔΔ−⋅Ct())(Tt((++mi))⋅−tT(t))]
startstart
∑
i=0
ρ ()m = (6.11)
R
n−1 n−1
2 2
[(Ct +⋅itΔΔ)(−⋅Ct)] [(Tt ++()mi ⋅−tT)(t))]
startstart
∑ ∑
i=0 i=0
The maximum cross correlation ρ is the maximum of all ρ ()m and ρ ()m . If ρ ()m and ρ ()m result
E L R L R
in the same value as maximum use the one which results from less time shifting steps. If time shifting is also
the same, prioritize ρ m . The number of the time shifting steps that yields the maximum cross correlation
()
L
ρ is defined as the phase error n . The corresponding shifted and truncated CAE curve C is recorded as
E ε
ts ts
C and the corresponding truncated test curve is recorded as T .
The phase score E is calculated by Formula (6.12). The best phase score is “1”, which means there is no
P
need to shift the CAE curve to reach the maximum cross correlation between the initial test and CAE curves.
*
If the time shift n is equal to or greater than the maximum allowable time shift threshold ε ⋅ N , then the
ε P
phase score is “0”. In between, the phase score is calculated by a regression method. It is either linear (
k =1 ), quadratic ( k =2 ), or cubical ( k =3 ).
P P P
10if n =

ε

k
P
*

 
ε ⋅−Nn

P ε
E = k ∈ 12,,3 (6.12)
  {}

P P
*
 
ε ⋅N

 P 

*
 0 if nN≥⋅ε

ε P
The pre-defined parameters shown in Table 6.3 are identical to the definition in Reference [1].
Table 6.3 — Fixed parameters of the phase score
Parameter Value Description
k
1 Exponent factor for calculating the phase score
P
*
0,2 Maximum allowable percentage of time shift
ε
P
6.4.2 Magnitude score
The magnitude error is a measure of discrepancy in the amplitude of the two time histories. It is defined
as the difference in amplitude of the two time histories when there is no time lag between them. Before
calculating the magnitude error, the difference between the time histories caused by error in phase and
slope (topology) are minimized by using dynamic time warping (DTW).

The definition of DTW is based on the notion of warping path. Let d be the matrix nn× of pair-wise squared
ts ts
distances between samples of C and T . This matrix d is called the local cost matrix. The function used
to calculate the value for each cell of the matrix is called local cost function di(, j). It is shown in
Formula (6.13).

inf ,,  ij≥+ ceil()01*N


di , j = inf ,,  ji≥+ ceil 01*N (6.13)
() ()


ts ts

Ci( )) −Tj() , otherwise
()

Where ceil means ceiling function. To make sure dynamic warping is within a reasonable range, maximum
allowed warping window is defined as 10 % length of the shifted and truncated curve. Any local cost function
falls outside this area will be set as infinite value as following Figure 6.4 shows.
Key
1 window size
2 acceptable warping band
Figure 6.4 — Cost function for allowable DTW window
Once the local cost matrix is built, the algorithm finds the alignment path which runs through the low-cost
areas on the cumulated cost matrix. A warping path w [Formula (6.14)] is a sequence of k matrix cells
K
[Formula (6.15)].
ww=≤,,ww., nk≤−()21n (6.14)
Kk12
wi=[],1jl≤≤k (6.15)
ll l
The cost of the warping path shall meet the following three conditions:
— Boundary conditions
w =[]11, and wn=[],n , i.e. w starts in the lower left cell and ends in the upper right cell.
1 k
— Continuity
Given wi=[], j and wi=[], j , then, ii−≤ 1 and jj−≤ 1 . This ensures that the cells of
ll−−11 l−1 ll l ll−1 ll−1
the warping path are adjacent.
— Monotonicity
Given wi= , j and wi= , j , then, ii−≥0 and jj−≥0 , with at least one strict
[] []
ll−−11 l−1 ll l ll−1 ll−1
inequality. This enforces w to progress over time.
The DTW distance is recursively computed using a dynamic programming approach that fills the cells of a
cumulative cost matrix di[], j and recurrence relation [Formula (6.16)]. The given sequence of d is
tw twmin
mandatory if the values of the minima are the same. Then the DTW distance is evaluated as shown in
Formula (6.17).
di,,ji ==11j
()


di(),,jd+−ij 11i =
[]
 tw
di, j = { (6.16)
[]

tw
di(),,jd+−[ij1 ]] j=1
 tw

di, jd+ otherwise
()
 twmin
Where dd=−min ij11,,di,,jd− ij−−11,
()[] [] []
twmintwtwtw
Dd= []nn, (6.17)
TW tw
The warping path which has a minimal cost associated with alignment is called the optimal warping path. It
ts ts
is found by following the definition that every possible warping path between C and T should be tested
which could be computationally challenging due to the exponential growth of the number of optimal paths
ts ts
as the lengths of C and T grow linearly.
ts ts
Any warping path w defines an alignment between C and T and, consequently, a cost to align the two
K
ts ts
histories. DC , T is the minimum of such costs, i.e. the cost of the optimal warping path
()
TWopt
[Formula (6.18)].
ts ts
DC ,,Td=min ij (6.18)
()
()
TWopt ll )
(∑
ij, ∈w
[]
w ll K
K
Let i and j represent the index of warping path of CAE and test data. An optimal warping path index w is
l l

formed as shown in Formula (6.19) with k steps in the path. It starts with in= and jn= , then it records
 
k k
each time step from nn, to 11, . The algorithms can be expressed as shown in Formula (6.20).
[] []
T
ii . ii.
  
l 1
kk−1

w = 1≤≤lk (6.19)
 
jj . jj.

l 1
 
kk−1
[,ij−=11] 1

ll

[,11ji−=] 1
ll


[,ij−−11][di ,]j =dd
[]ij, = (6.20)
 ll tw ll twmin
ll−−11

[,ij −−11][di ,]jd=
ll tw ll twmin


[,ij−−11][di −−11, j 1]=d

ll tw ll twmin
Where dd=−min()[]ij11,,di[],,jd− []ij−−11, . Hence, the index matrix of warping path w
twmintwtwtw
ts ts
can be expressed by the index of CAE ( C ) and test (T ) curves.
ts+w ts+w
Then the truncated and warped CAE curve C and the shifted, truncated, and warped test curve T
are formed as shown in Formulae (6.21) and (6.22).

ts++wtsw ts+wtsts ts++wtsw k
   
Ci ,,Ci .,Ci() = Cn(),,Cn()−1.,CC()1 ∈R (6.21)
() ()

kk−1
   

ts++wtsw ts+wtsts ts++wtsw k
   
Tj ,,Tj .,Tj = Tn ,,Tn−1.,TT1 ∈R (6.22)
() () () () () ()

kk−1
   
The magnitude error ε is calculated by Formula (6.23).
mag
ts++wtsw
CT−
ε = (6.23)
mag
ts+w
T
*
Formula (6.24) is used to calculate the magnitude score E , where ε is the maximum allowable magnitude
M M
error and k defines the order of the regression. The best magnitude score is “1”, which means there is no
M
difference in the amplitudes after phase shift and dynamic time warping. If the magnitude error ε is
mag
*
equal to or greater than the maximum allowable magnitude error threshold ε , then the magnitude score is
M
“0”. In between, the magnitude score is calculated by regression method.
 10if ε =
mag

k
M
 *
 
εε−
 Mmag
 
Ek= ∈ 1,223, (6.24)
{}

M M
*
 
ε

M
 

*

0 if εε≥
 magM
The pre-defined parameters shown in Table 6.4 are identical to the definition in Reference [1].
Table 6.4 — Fixed parameters of the magnitude score
Parameter Value Description
k 1 Exponent factor for calculating the magnitude score
M
*
0,5 Maximum allowable magnitude error
ε
M
6.4.3 Slope score
The slope error is a measure of discrepancy in slope (topology) of the two time histories. The slope of a time
history is defined by the slope at each point. To ensure that the effect of global time shift is minimized, the
ts ts
slope is calculated from the time shifted histories T and C .

Central differences are first used to calculate the slope, except the first point taking a forward difference
and the last point taking a backward difference.
ts ts

Ci()+11−Ci() /Δtiif =
()


ts+d ts ts
C = Ci()+11−−Ci() /21Δ ()

ts ts

Ci()−−Ci()1/Δtiif =n
()

ts ts

Ti()+11−Ti() /Δtiif =
()


ts+d ts ts
T = Ti()+11−−Ti() /21Δ ()

ts ts

Ti()−−Ti()1/Δtiif =n
()

Next, the average slope is calculated with a continuous nine-point interval to generate smoother slope
ts+d ts+d
curves ( C and T ).
ts+d

Ci()   if in= 1,

i+11

ts+d

Ck()  if in=−21,
∑ 0

ki=−1

i+2

ts+d
 Ck() if in=−32,
 ∑ 0
ts+d
C = 5 (6.27)

ki=−2

i+3

ts+d
Ck() if i==−43,n

∑

ki=−3

i+4

ts+d
Ck() if4< 
∑ 0


ki=−4
ts+d

Ti()  if in= 1,

i+1

ts+d

Tk()  if in=−21,
∑ 0

ki=−1

i+2

ts+d
 Tk() if in=−32,
 ∑ 0
ts+d
T = 5 (6.28)

ki=−2

i+3

ts+d
Tk() if in= 4, −−3

∑

ki=−3

i+4

ts+d
Tk() if4< 
∑ 0


ki=−4
Therefore, the slope curves are in same length with time shifted history curves and are used to calculate the
slope error directly without performing dynamic time warping. Both curves are then used to calculate the
slope error ε by Formula (6.29).
slope
ts++dtsd
CT−
ε = (6.29)
slope
ts+d
T
*
Formula (6.30) is used to calculate the slope score E , where ε is the maximum allowable slope error and
S S
k defines the order of the regression. The best slope score is “1”, which means there is no difference
S
between the two curve’s slope. If the slope error ε is equal to or greater than the maximum allowable
slope
*
slope error ε , then the slope score is “0”. In between, the slope score is calculated by regression method.
S
 10if ε =
slope

k
S
 *
 
εε−
 Sslope
 
Ek= ∈ 12,,3 (6.30)
{{}

S S
*
 
ε

S
 

*

0 if εε≥
slopeS

The pre-defined parameters shown in Table 6.5 are identical to the definition in Reference [1].
Table 6.5 — Fixed parameters of the slope score
Parameter Value Description
k 1 Exponent factor for calculating the slope score
S
*
2,0 Maximum allowable slope error
ε
S
6.4.4 Step by step procedure
The following step by step process shall be followed to calculate the phase, magnitude and slope sub-ratings.
a) Pre-process both signals according to Clause 8 ( C and T ).
b) Calculate the phase error in terms of time steps n by maximizing cross correlation.
ε
c) Calculate the phase score E .
P
ts ts
d) Calculate the shifted and truncated time history curves C and T .
e) Perform dynamic time warping to the shifted and truncated time history curves to generate the shifted,
ts+w ts+w
truncated and warped time history curves C and T .
ts+w ts+w
f) Calculate the magnitude error ε between C and T .
mag
g) Calculate the magnitude score E .
M
ts+d ts+d
h) Generate the shifted and truncated derivative time history curves C and T .
0 0
ts+d ts+d
i) Generate the smoother shifted and truncated derivative time history curves C and T by
averaging nine-point intervals.
ts+d ts+d
j) Calculate the slope error ε between C and T .
slope
k) Calculate the slope score E .
S
7 Meaning of the overall ISO rating
The objective rating R ranges from “0” to “1”. The higher the score the better the correlation of the two
signals. This single-rating number can be transferred to a grade that represents the level of the correlation
by using a sliding scale (see Table 7.1).

Table 7.1 — Sliding scale of the overall ISO rating
Rank r
Rating R
Grade Description
Almost perfect characteristics of the reference signal
R >0,94
1 Excellent
is captured
Reasonably good characteristics of the reference sig-
2 Good 0,80< R ≤0,94 nal is captured, but there are noticeable differences
between both signals
Basic characteristics of the reference signal is cap-
0,58< R ≤0,80
3 Fair tured but there are significant differences between
the two signals
4 Poor R ≤0,58 Almost no correlation between the two signals
T
...