# Developing Synthetic Spectroscopy Noise and Chemometric Database for
Computational Classification

Nicholas J. Napoli
Dept. of Electrical and Computer Engineering
University of Florida
Gainesville, FL 32611
<EMAIL_ADDRESS>
Please request code of the the noise generator or additional documentation

###### Abstract

There has been little to no work in the area of spectroscopy noise in order to
create data sets for analytical algorithms to be challenged on the ability to
separate chemicals. We present a framework on how to build off of a sparse
about of experimental data in order to expand your chemometric database and
create realistic instrumentation noise. The combination of various
interactions of chemicals combined with various random permutations of
spectroscopy noises enables researches to better capture and model the
multitude types of signals and variations that can be present within an
experimental reading.

##  1 Introduction

The unavailability of open source spectroscopy data and the difficulty in
direct acquisition of spectroscopy data via experiments are significant
hurdles in the development of effective detection algorithms associated with
such data in chemometrics. To address this issue, we sought to develop a
method to generate a synthetic spectroscopy data bank which attempts to
faithfully capture the operationally relevant features of fluorescent spectra.
We wanted this method to possess the ability to account for the variations
arising from different excitation wavelengths, multiple combinations of
chemical mixtures, different apparatus and effects of user error.

##  2 Data from PhotoChemCAD

To generate such a synthetic database, we started with a sparse amount of
reputable data from PhotoChemCAD [1, 2]. To achieve our objective with this
limited set of experimental data, we made several assumptions:

  * •

There is no chemical quenching.

  * •

The spectral resolution is accurately depicted through interpolation and
decimation.

  * •

All path lengths are $1\,cm$.

  * •

Accurate depiction of the molar absorptivity is projected by scaling the
absorption spectrum by cited molar extinction coefficients.

  * •

Beer-Lambert’s law is obeyed.

  * •

Concentrations are below $1\mu M$ to minimized inner filter effects to achieve
additive absorption and emission spectra.

From PhotoChemCAD, we chose multiple chemicals within the same solvent,
Toluene. The analytes selected were from two chemical classes: oligopyrrole
and polycyclic aromatic hydrocarbons. These two criteria were selected to
obtain data that are similar in chemical composition. The resulting strong
spectral similarities create a “non-trivial” data set for classification and
quantification purposes. The chemicals that were used in the database are
listed in Table 1 and 2.

The data from PhotoChemCAD provide neither the concentration of the analytes
nor the emission characteristics at different excitation wavelengths. In our
data set, we acquire these two quantities directly from published results for
spectral modeling see Table 1 and 2. Epsilon, $\varepsilon$, enables us to
account for different concentrations of the chemical; and the quantum yield,
$\Phi_{F}$, enables us to calculate different emissions spectra at various
excitation wavelengths. With these quantities accounted for, we are able to
generate a more realistic data set.

Chemical | Solvent |  Epsilon ($\varepsilon$) in $cm^{-1}/M$ at $\lambda_{Ex}$ |  Quantum Yield ($\Phi_{F}$) | Cited
---|---|---|---|---
5,10-Diaryl Chlorin | Toluene |  $89,100$ at $414\,nm$ |  $0.260$ | [3]
5,10-Diaryl Mg-oxoChlorin | Toluene |  $191,000$ at $408\,nm$ |  $0.100$ | [3]
5,10-Diaryl oxoChlorin | Toluene |  $174,000$ at $414\,nm$ |  $0.130$ | [3]
5,10-Diaryl Zn-Chlorin | Toluene |  $186,000$ at $412\,nm$ |  $0.083$ | [4, 3]
5,10-Diaryl Zn-oxoChlorin | Toluene |  $209,000$ at $408\,nm$ |  $0.040$ | [3]
Bis(5-mesityldiprinato)zinc | Toluene |  $115,000$ at $487\,nm$ |  $0.360$ | [5]
Bis(5-phenyldiprinato)zinc | Toluene |  $115,000$ at $485\,nm$ |  $0.006$ | [5]
Magnesium Octaethylporphyrin | Toluene |  $408,300$ at $410\,nm$ |  $0.150$ | [6, 7]
Magnesium Tetramesityporphyrin | Toluene |  $446,700$ at $426.5\,nm$ |  $0.170$ | [8, 7]
Magnesium Tetraphenylporphyrin | Toluene |  $22,000$ at $564\,nm$ |  $0.150$ | [9, 10]
Table 1: Chemicals in the Database: Oligopyrrole. Chemical | Solvent |  Epsilon ($\varepsilon$) in $cm^{-1}/M$ at $\lambda_{Ex}$ |  Quantum Yield ($\Phi_{F}$) | Cited
---|---|---|---|---
Perylene-diimide | Toluene |  $44,000$ at $490\,nm$ |  $0.97$ | [11]
Perylene-Monoimide | Toluene |  $32,000$ at $511\,nm$ |  $0.86$ | [12]
Perylene-Monoimide(OR)3 | Toluene |  $32,000$ at $479\,nm$ |  $0.91$ | [12]
Perylene-Monoimide (OR) | Toluene |  $40,000$ at $507\,nm$ |  $0.82$ | [12]
Table 2: Database Chemicals: Polycyclic Aromatic Hydrocarbons.

###  2.1 Preprocessing of Spectral Data

The spectral data from PhotoChemCAD is an agglomeration of various sources,
where acquisition parameters are distinctive from each other. The spectral
data from the selected chemicals are different in their wavelength range and
optical resolution. To ensure proper manipulation among different chemical
spectral vectors, the length and indexing of the vectors are required to be
equivalent. The optical sampling of the spectral vectors obtained were either
$0.25\,nm$, $0.5\,nm$, or $1.0\,nm$; hence, the vectors were interpolated or
decimated to get values corresponding to $0.5\,nm$. Each vector was padded
with elements of value $1.0\cdot 10^{-20}$ to provide a uniform wavelength
range throughout the entire database of selected chemicals. The value
$1.0\cdot 10^{-20}$ is used to avoid absolute zero errors and to circumvent
subsequent complications with vector and matrix manipulations.

###  2.2 Generating Spectra Corresponding to Various Concentrations

A spectrometer takes measurements of light absorption, producing a unique
spectral absorption signature. When taking measurements, the concentration and
pathlength are held constant. The epsilon value, which is a function of the
excitation wavelength, is an intrinsic property of the measured chemical that
defines the spectral waveform characteristics. The selected data only provides
the absorbance, giving no insight into the concentration or pathlength. We are
therefore unable to distinguish the epsilon values due to the unknown
collection parameters from the various sources provided by PhotoChemCAD. The
measured absorbance of the sample is proportional to the number of absorbing
molecules from the incident light of the spectrometer and it is essential that
the absorbance value is corrected for a meaningful comparison [13]. This
correction for absorption is referred to as _molar absorptivity_ or _molar
extinction coefficients_ , which serves to compare spectra and evaluate the
relative strength of the absorbance. In order perform a proper comparison
between spectra, we scale the spectral vector with respect to epsilon at its
appropriate listed excitation wavelength from Table 1 and 2.

Consider a measured absorption data vector from Table 1:

$\textbf{D}_{Ab_{j}}=\begin{bmatrix}d_{Ab_{j1}}&d_{Ab_{j2}}&\cdots&d_{Ab_{jN}}\end{bmatrix}^{T},$
(1)

where $d_{Ab_{jk}}\in[0,A],\forall k\in\overline{1,N}$, $A$ is an arbitrary
positive number, and $j$ is a specific chemical. These optical absorption
measurements were scaled to coincide with cited molar extinction coefficients
(i.e., epsilon) at the corresponding wavelength from Table 1 and 2 via the
following equation:

$\textbf{S}_{Ab_{j}}=\textbf{D}_{Ab_{j}}\frac{\varepsilon_{\lambda_{Ex}j}}{D_{Ab_{j}\lambda{{}_{Ex}}}}C,$
(2)

where $\varepsilon_{\lambda_{Ex}j}$ is the molar extinction coefficient of
chemical $j$ from Table 1 and 2, $C$ is the concentration, and
$D_{Ab_{j}\lambda_{Ex}}$ is the element in $\textbf{D}_{Ab_{j}}$ that is
associated with epsilon at the specific excitation wavelength of
$\lambda_{Ex}$. With the spectral vector properly scaled, we assume that each
spectral element depicts its appropriate molar extinction coefficient for all
wavelengths. We can now apply Beer-Lambert’s law to expand the database by
altering the concentration while we hold constant the pathlength at $1\,cm$.

###  2.3 Expanding the Data Set Via Quantum Yield

When a molecule is excited to a higher quantum state of a particle and it
transitions to a lower state, the molecule emits a photon. The more the
molecule absorbs energy the higher potential for it to elicit more photons.
The amount of fluorescence emission is a function of the amount of light
absorbed by a molecule. This function is known as the _quantum yield of the
fluorescence, $\Phi_{F}$._ It is defined as the number of photons emitted over
the total number of photons absorbed [14]. Based on the intrinsic nature of
the molecule and its absorption properties, specific wavelengths are more
prone to be absorbed than others. It is apparent that the excitation
wavelength affects the total intensity of the absorption and concurrently
affects the total emission intensity. By dynamically changing the excitation
wavelength, we generate various emission spectra accounting for the effect of
quantum yield.

Consider a measured fluorescence emission spectrum vector from Table 1 and 2:

$\textbf{D}_{Em_{j}}=\begin{bmatrix}d_{em_{j1}}&d_{em_{j2}}&\cdots&d_{em_{jN}}\end{bmatrix}^{T},$
(3)

where $D_{em_{jk}}\in[0,A],\forall k\in\overline{1,N}$ , $A$ is an arbitrary
positive number, and $j$ is a specific chemical. In order to generate further
data and for a realistic simulation of different excitation wavelengths, we
consider $I_{o\lambda}$ as the intensity of incident light to excite the
sample. We are able to quantify the summed intensity of the emission
fluorescence, $S_{Em}$, as

$\textbf{S}_{Em}\propto\textbf{N}_{Em_{j}}S_{Ab_{j}\lambda_{\varepsilon}}\Phi_{F_{j}}I_{o\lambda},$
(4)

where $S_{Ab_{j}\lambda_{\varepsilon}}$ is the element in
$\textbf{S}_{Ab_{j}}$ that is associated with epsilon at $\lambda_{Ex}$ and
$\textbf{N}_{Em_{j}}$ is the normalized vector of $\textbf{S}_{Em_{j}}$, i.e.,

$\textbf{N}_{Em_{j}}=\frac{\textbf{D}_{Em_{j}}}{\displaystyle\sum_{k=1}^{N}\textbf{D}_{Em_{j}}}.$
(5)

Note that the summed fluorescence emission is dependent on the incident light
intensity, the absorbance magnitude at a particular $\lambda_{Ex}$, and
$\Phi_{F}$ [14]. As shown in Figure 1, different $\lambda_{Ex}$ s obtained
from the absorption spectrum elicit different energy contributions to the
spectral topology of the emission signal.

Figure 1: Emissions as a Function of $\lambda$ and $\Phi_{F}$.

###  2.4 Generating Spectra Corresponding to Combinations of Chemicals

Contingent on our assumption that the absorption and emission spectra are
additive, we expand our database to include linear combinations of chemicals.
This assumption is valid only when Beer-Lambert’s law is obeyed and the inner
filter effect is minimized, thus allowing us to define the linear combination
process at the wavelength of interest as

$A^{\lambda_{1}}_{x+y}=A^{\lambda_{1}}_{x}+A^{\lambda_{1}}_{y}=\epsilon^{\lambda_{1}}_{x}bC_{x}+\epsilon^{\lambda_{1}}_{y}bC_{y}.$
(6)

One way to introduce an amount of different chemicals is by sampling without
replacement and without ordering. This can be accomplished via 7 and expressed
as a binomial coefficient, where $k$ is the number of analytes that are chosen
from a set of $n$ total number of analytes [15]:

$C^{n}_{k}=\frac{n(n-1)\ldots(n-k+1)}{k!}=\frac{n!}{k!(n-k)!}=\left(\begin{matrix}n\\\
k\end{matrix}\right)$ (7)

However, our intention is to evaluate these analytes over different
combinations of $k$. A visual example of four different chemicals is shown in
Table 3.

$k=1$ | $k=2$ | $k=3$ | $k=4$
---|---|---|---
Chem(1) | Chem(1,4) | Chem(1,2,3) | Chem(1,2,3,4)
Chem(2) | Chem(4,2) | Chem(1,2,4) |
Chem(3) | Chem(4,3) | Chem(1,3,4) |
Chem(4) | Chem(3,1) | Chem(2,3,4) |
| Chem(3,2) |  |
| Chem(2,1) |  |
Total = 4 | Total = 6 | Total = 4 | Total = 1
Table 3: Evaluating Chemical Permutations.

Hence, we need to define equation 7 over a sum of all $k$:

$\sum^{k=n}_{k=1}\frac{n!}{k!(n-k)!}=\sum^{k=n}_{k=1}\left(\begin{matrix}n\\\
k\end{matrix}\right).$ (8)

It is apparent that, as we evaluate various $n$ number of sets using equation
8, we can get $2^{n}-1$ different chemical combinations (the $-1$ is due to
the fact that we are not considering the scenario where there is no chemical
present within the database). These combinational permutations can also be
later leveraged by utilizing probability detection paradigms and matching
algorithms to the classification of spectroscopy signal [16, 17],

For software implementation purpose, we use a binary representation for
chemical presence/absence within the sample. This is illustrated by amending
Table 3 with the appropriate coded binary representation, where each bit
represents chemical presence/absence ($1$=Chemical Present and $0$=Chemical
Not Present).

k=1 | k=2 | k=3 | k=4
---|---|---|---
Chem(1)=[0001] | Chem(1,4)=[1001] | Chem(1,2,3)=[0111] | Chem(1,2,3,4)=[1111]
Chem(2)=[0010] | Chem(4,2)=[1010] | Chem(1,2,4)=[1011] |
Chem(3)=[0100] | Chem(4,3)=[1100] | Chem(1,3,4)=[1101] |
Chem(4)=[1000] | Chem(3,1)=[0101] | Chem(2,3,4)=[1110] |
| Chem(3,2)=[0110] |  |
| Chem(2,1)=[0011] |  |
Table 4: Binary Coding of Chemicals.

##  3 Errors in Spectroscopy Measurements

Spectroscopy signals are also affected by electronic noise, stray light, light
scattering, wavelength accuracy, resolution, stability, baseline flatness,
effects of sampling geometry, and user error [18]. While it is not realistic
to accommodate all these types of errors, we now discuss how several
additional sources of error are introduced into our synthetic spectroscopy
data set.

###  3.1 User Error

Use error is quite common in spectroscopy data. The most common user error
involves a lack of concentration, usually associated with pipetting chemical
dilutions at low concentrations. This error can be further exacerbated if the
molar absorptivity is high. Applying Beer-Lambert’s law to this error, one
would expect changes in peak height and overall spectral area. Another user
error involves fluorescent contamination of the measured sample, or when the
detected light is contaminated by Rayleigh or Raman scatter. This is also
contingent on the particle size of the analyte, which is a function of the
variance within the measured spectrum [19]. Figure 2 shows how the emission
remains the same with different excitation wavelengths in a pure sample, and
how a contaminate alters the emission spectrum topology at $420\,nm$ [18].

Figure 2: Emission Spectra of $C102$ and a Mixture of $C102$ with the
Fluorescent Impurity $C153$ [18].

###  3.2 Stray Light

Stray light is the measured light of any wavelength reaching the detector that
is not associated with the bandwidth of the selected wavelength [20]. Stray
light manifests itself as an apparent deviation in Beer-Lambert law. The
effects of stray light is a decrease in absorbance and a reduction of the
perceived projected linearity of the absorbance. This can be described by
equation 9, where $I$ is the transmitted light, $I_{s}$ is the stray light,
and $I_{o}$ is the incident light:

$Absorbance=-\log\left(\frac{I+I_{s}}{I_{o}+I_{s}}\right).$ (9)

Figure 3 shows the effect of stray light on the absorbance [20].

Figure 3: Apparent Absorbance vs True Absorbance with Increasing Stray Light
[20].

###  3.3 Wavelength Accuracy

Wavelength accuracy is the inability to preserve the wavelength scaling at the
detector or emitter. This scaling error introduces a shift in the measured
wavelength. This causes our perception of the true $\lambda_{max}$ to be
inaccurate [18, 20]. See Figure 4.

Figure 4: Wavelength Accuracy [18].

###  3.4 Self-Absorption

Self-absorption depends upon the geometric arrangement observing the
fluorescence and high optical densities, which can cause intensity distortion
within specific wavelength ranges. As can be seen in Figure 4, the error
causes a shifting of the spectrum. Figure 5 is an example of a right-angle
observation, where short wavelength emissions are attenuated by the analyte
Anthracence’s absorbance at the shorter wavelengths [18].

Figure 5: Effects of Self-Absorption.

###  3.5 Overview of Spectroscopy Errors

From the discussion above, one notices that a large proportion of the errors
associated with spectroscopy measurements cause the nominal spectrum to be
perturbed in a “smoother” manner. Other types of errors that generate higher
frequency perturbations in the spectroscopy measurements are typically modeled
as additive white gaussian noise (AWGN). For the current purpose, our
intention is to model the errors that have a higher potential to elicit a
misclassification within the database. This is what we undertake in the
following sections.

##  4 Insertion of Spectroscopy Measurement Perturbations

The introduction of perturbations to the database allows for the creation of
more realistic test samples in our prototype data set. This enables us to
explore how signal degradation can affect classification performance of our
algorithm.

###  4.1 Windowing

We employ different windowing functions as the basic strategy to alter the
spectral emission and absorption vectors. The windowing foundation is only the
basis of defining, where the types of window functions will be implemented in
the spectra. These locations for each type of windowing function shares a
relationship with the spectral peaks of emission and absorption vectors. In
later sections, the windowing functions are modified to provided either
compression or dilation to the spectra within a specified design range.

Consider the following spectral data vector (corresponding to emission or
absorption):

$\textbf{S}=\begin{bmatrix}S_{1}&S_{2}&\cdots&S_{N}\end{bmatrix}^{T},$ (10)

where $S_{k}\in[0,A],\,\forall k\in\overline{1,N}$, and $A$ is an arbitrary
real positive constant. The peaks of this vector were examined to adaptively
create perturbations in the vicinity of these peaks. This was done to create a
unique correlated noise (Dilation or Compression) to individual spectral
vector. This correlated noise is dependent on S’s spectral peak “shape”, the
windowing function type ($t$), and the given window size $L_{w}$, where
$L_{w}\ll N$. The peaks were sought by implementing the Matlab function
findpeaks, where we set a minimum window distance $L_{w}$ between peaks. The
findpeaks function yields a vector P, where the maximum peak location for the
processed spectra are represent by the elements of P:

$\textbf{P}=[k_{1},\hskip 14.22636ptk_{2},\hskip 5.69054pt\cdots k_{K}]^{T},$
(11)

where $k_{p}\in[1,N],\forall p\in\overline{1,K},$ and $K\leq N/L_{w}$. Given
$L_{w}$ and $N$, we determine the number of windows that will be designed by
taking the integer quotient $\lfloor N/L_{w}\rfloor=C$, where $C$ is the
number of windows to be designed. This yields $C+1$, window segments. We
determine the type of windowing function that will be employed to each window
segment, $i$, based on our _peak indictor_ vector, $\textbf{I}_{p}$. The peak
indictor vector informs us at what window segment a peaks occur by taking the
integer quotient plus a unit ones vector $\textbf{U}_{K}$, where $K$ is the
length.

$\textbf{I}_{p}=\lfloor\textbf{P}/L_{w}\rfloor+\textbf{U}_{K}$ (12)

We use vector $\textbf{I}_{p}$ to determine, $t$, the type of windowing
function to implement for $\textbf{W}_{i,t}$. Each $\textbf{W}_{i,t}$ is
characterized with one of five possible windowing type functions, $t$, on the
$i^{th}$ window segment, where $\forall i\in\overline{1,C+1}$ . The $i^{th}$
window segment is associated to S’s spectral data by $[S_{1+(i-1)\cdot
L_{w}},S_{i\cdot L_{w}(i-1)+L_{w}}]$.

$\textbf{W}_{i,t}=[W_{1},\hskip 14.22636ptW_{2},\hskip 5.69054pt\cdots\hskip
5.69054ptW_{L_{w}}]^{T},$ (13)

where, $W_{j}\in[0,1],\forall j\in\overline{1,L_{w}},\forall
t\in\overline{1,5},\forall i\in\overline{1,C}$. Equation 14, handles the
residual data of $S$, since we only designed $C$ windows.

$\textbf{W}_{(C+1),t}=[W_{1},\hskip 14.22636ptW_{2},\hskip
5.69054pt\cdots\hskip 5.69054ptW_{N-(L_{w}\cdot C)}]^{T},$ (14)

where, $W_{j}\in[0,1],\forall j\in\overline{1,L_{w}},\forall
t\in\overline{1,5},\forall i\in\overline{1,C}$. The type of windowing function
that is implemented for each $i$ window segment is determined by recursively
examining each case in numerical order until the specific conditions of a case
is in accordance of the criteria. The windowing functions and conditioned
criteria are defined by the following five cases, where for all cases
$\alpha\in[0,1]$, $\gamma=(L_{w}-1)$, and $\beta=(1-\frac{\alpha}{2})$.

_Windowing Case 1: Hanning Window_

$\textbf{W}_{i,1}(j)=.5(1-\cos(2\pi(\frac{j}{L_{w}})))$ (15)

$\mbox{{if and only if}}\hskip 2.84526pt\exists i\in\textbf{I}_{p}$

_Windowing Case 2: Tukey Window_

$\textbf{W}_{i,2}(j)=\left\\{\begin{array}[]{ll}\frac{1+\cos{(\pi(\frac{2(j-1)}{\alpha(\gamma)})-1)}}{2},&\mbox{for
$1\leqslant j\leqslant\frac{\alpha(\gamma)}{2}$}\\\ 1,&\mbox{for
$\frac{\alpha(\gamma)}{2}\leqslant j\leqslant(\gamma)(\beta)$}\\\
\frac{1+\cos{(\pi(\frac{2(j-1)}{\alpha(\gamma)})-\frac{2}{\alpha}+1)}}{2},&\mbox{for
$(\gamma)(\beta)\leqslant j\leqslant(\gamma)$}\\\ \end{array}\right.$ (16)

$\mbox{{if and only if}}\hskip 2.84526pt\exists
i\not\in\textbf{I}_{p}\wedge\exists(i-1)\in\textbf{I}_{p}\wedge\exists(i+1)\in\textbf{I}_{p}$

_Windowing Case 3: Modified Left Tukey Window_

$\textbf{W}_{i,3}(j)=\left\\{\begin{array}[]{ll}\frac{1+\cos{(\pi(\frac{2(j-1)}{\alpha(\gamma)})-1)}}{2},&\mbox{for
$1\leqslant j\leqslant\frac{\alpha(\gamma)}{2}$}\\\ 1,&\mbox{for
$\frac{\alpha(\gamma)}{2}\leqslant j\leqslant(\gamma)$}\\\ \end{array}\right.$
(17)

$\mbox{{if and only if}}\hskip 2.84526pt\exists
i\not\in\textbf{I}_{p}\wedge\exists(i-1)\in\textbf{I}_{p}\wedge\exists(i+1)\not\in\textbf{I}_{p}$

_Windowing Case 4: Modified Right Tukey Window_

$\textbf{W}_{i,4}(j)=\left\\{\begin{array}[]{ll}1,&\mbox{for $1\leqslant
j\leqslant(\gamma)(\beta)$}\\\
\frac{1+cos{(\pi(\frac{2(j-1)}{\alpha(\gamma)})-\frac{2}{\alpha}+1)}}{2},&\mbox{for
$(\gamma)(\beta)\leqslant j\leqslant(\gamma)$}\\\ \end{array}\right.$ (18)

$\mbox{{if and only if}}\hskip 2.84526pt\exists
i\not\in\textbf{I}_{p}\wedge\not\exists(i-1)\in\textbf{I}_{p}\wedge\exists(i+1)\in\textbf{I}_{p}$

_Windowing Case 5: Null Variance Window_

$\textbf{W}_{i,5}(j)=1,\hskip 34.1433pt\mbox{for $\forall j$}\\\ $ (19)

$\mbox{{if and only if}}\hskip 2.84526pt\exists
i\not\in\textbf{I}_{p}\wedge\exists(i-1)\not\in\textbf{I}_{p}\wedge\exists(i+1)\not\in\textbf{I}_{p}$

Each individual $\textbf{W}_{i,t}$ window vector that is designed will be
cascaded in numerical order to construct, $\textbf{J}_{F}$, the foundation for
creating our dilation compression vector to modify vector S. Hence,
$\textbf{J}_{F}$ is defined as:

$\textbf{J}_{F}=[\textbf{W}_{1,t},\hskip 5.69054pt\textbf{W}_{2,t},\hskip
14.22636pt\cdots\hskip 5.69054pt\textbf{W}_{C,t},\hskip
5.69054pt\textbf{W}_{(C+1),5}]^{T},$ (20)

where $\textbf{W}_{i,t}\in[0,A],\forall i\in\overline{1,C},\forall
t\in\overline{1,5},$ and $\textbf{J}\in[0,A],\forall i\in\overline{1,(C+1)}$.
However, note we only design $C$ windows, the (C+1) window will always default
to case $5$ to handle the residual data of S thats smaller than the specified
window size, $L_{w}$.

###  4.2 Basic Design for Dilation Noise

Once our foundation vector is set, we modify, $W_{i,t}$. We demonstrate a few
different methods to modify $W_{i,t}$ to create the most ideal synthetic noise
representation starting from the most basic. As well, we review the down falls
in order to improve upon each method. We first exemplify synthetic noise to
the data by simply dilating the window locations where the peaks occur. This
dilation is scaled by a constant $\eta$. Therefore, we revamp $W_{1,t}$, to be
scaled by $\eta$ and force other Tukey windows $W_{2-4,t}$ to one:

$\displaystyle\hat{\textbf{W}}_{\hat{D}_{1,t}}=(\eta)\cdot\textbf{W}_{1,t}+U_{L_{w}},$
(21)
$\displaystyle\hat{\textbf{W}}_{\hat{D}_{2-5,t}}=(0)\cdot\textbf{W}_{2-5,t}+U_{L_{w}},$
(22)
$\displaystyle\textbf{J}_{\hat{D}}=[\hat{\textbf{W}}_{\hat{D}_{1,t}},\hskip
5.69054pt\hat{\textbf{W}}_{\hat{D}_{2,t}},\hskip 14.22636pt\cdots\hskip
5.69054pt\hat{\textbf{W}}_{\hat{D}_{C,t}},\hskip
5.69054pt\textbf{W}_{(C+1),5}]^{T},$ (23)
$\displaystyle\textbf{S}_{S+N}=\textbf{J}_{\hat{D}}\hskip
2.84526ptdiag(\textbf{S})$ (24)

Figure 6: Absorbance Spectrum with Dilation Noise.

We can see in figure 6, that by implementing this method we are only given the
option of dilating the spectrum at a constant $\eta$. In addition, its worth
noting due to the fixed windowing segments, the alignment of the peaks are not
centered directly over the windowed segment causing shifting of the peak
wavelength. In some scenarios, this maybe considered ideal for further
perturbations of the signal, where the shifting of the peak wavelength is a
function of $\eta$.

###  4.3 Basic Design for Compression Noise

Given that we can dilate the signal, in order for us to compress the signal, a
compression vector is designed as such by equation 25, 26, and 27, which is
pictorially represented in Figure 8:

$\displaystyle\hat{\textbf{W}}_{\hat{C}_{i,t}}=\eta\cdot\textbf{W}_{i,t}+(U_{L_{w}}-\eta
U_{L_{w}})$ (25)
$\displaystyle\textbf{J}_{\hat{C}}=[\hat{\textbf{W}}_{\hat{C}_{1,t}},\hskip
5.69054pt\hat{\textbf{W}}_{\hat{C}_{2,t}},\hskip 14.22636pt\cdots\hskip
5.69054pt\hat{\textbf{W}}_{\hat{C}_{C,t}},\hskip
5.69054pt\textbf{W}_{(C+1),5}]^{T},$ (26)
$\displaystyle\textbf{S}_{S+N}=\textbf{J}_{\hat{C}}\hskip
2.84526pt\textit{diag}(\textbf{S})$ (27)

where $D_{k}\in[0,A],\forall k\in\overline{1,N}$

Figure 7: Absorbance Spectrum with Compression Noise.

###  4.4 An Axiomatic Approach for Designing Appropriate Noise

In the previous sections, we discussed the foundation of the windowing and the
functions that were integral for dilation and compression of the signal by a
factor of $\eta$. It’s only sensible to anticipate the above methods as a
combination of random dilation and compression for a more convincing simulated
noise. Considering these two functions as a whole, comprised of different
adjacent windowing techniques, it is careless to assume that $\eta$’s effect
on the signal is equivalent. A pragmatic approach of declaring guidelines for
the noise algorithm must be established to meet the appropriate standards to
have unbiased detection simulations when noise is introduced. In order to
fortify this concept of why guidelines are required and to exam what
guidelines that need to be put in place, we introduce an algorithm that
functionally fails as a dilating compressing algorithm.

####  4.4.1 Failure for Proper Dilation and Compression

$\displaystyle\hat{\textbf{W}}_{i,1}=\eta
X\textbf{W}_{i,1}+(1-\frac{\eta}{2})U_{L_{w}},$ (28)
$\displaystyle\hat{\textbf{W}}_{i,2-4}=(\eta)\textbf{W}_{i,2-4}+(1-\frac{\eta}{2})U_{L_{w}}),$
(29) $\displaystyle\hat{\textbf{W}}_{i,5}=\textbf{W}_{i,5},,$ (30)
$\displaystyle\textbf{J}_{\hat{CD}}=[\hat{\textbf{W}}_{\hat{D}_{1,t}},\hskip
5.69054pt\hat{\textbf{W}}_{\hat{D}_{2,t}},\hskip 14.22636pt\cdots\hskip
5.69054pt\hat{\textbf{W}}_{\hat{D}_{C,t}},\hskip
5.69054pt\textbf{W}_{(C+1),5}]^{T},$ (31)
$\displaystyle\textbf{S}_{S+N}=\textbf{J}_{\hat{CD}}\hskip
2.84526pt\textit{diag}(\textbf{S})$ (32)

where X is a random variable uniformly distributed $X\in[0,1]$

Figure 8: Absorbance Spectrum with Noise Failing to Dilate.

Visually examining the noise vector, we can note that the hanning windows
exceeds the peak magnitude of one. We may assume the data vector will undergo
dilation and compression, but this is deceiving. When the filter is applied to
the data vector the data is not properly dilated as expected. It begins to
become apparent, graphically in Figure 8 that, the magnitude of dilation on
the sample may not be comparable to the compression. However, the improper
dilation may be attributed to that particular sample, therefore simulations
are done in the preceding section to examine how the intensity changes are
distributed across numerous samples.

####  4.4.2 Noise Design Guidelines

The previous example shows the possible cause of how the noise is distributed
over the vector. When examined over numerous trials, it is possible that it
may not be ideal for testing our detection algorithm in the later chapters.
Hence, we set guidelines of what our ideal noise distribution should be in
order to achieve an appropriate testing scenario for our detection algorithm.
The guidelines are verified by visual inspection from simulations that produce
the noise distribution.

_Synthetic Noise Distribution Ideal Guidelines_

  1. 1.

$\eta$ will be random for each designed window.

  2. 2.

Within a single noise vector, Dilation and Compression can occur at various
locations.

  3. 3.

As $\eta$ increases, the variance of the distribution will increase.

  4. 4.

The noise distribution when evaluating the magnitude between corresponding
elements should be symmetric around zero.

  5. 5.

The distribution of intensity loss and gain for a entire vector should be
symmetric around zero.

These guidelines will assist in the development of a pragmatic noise that will
better challenge the subsequent detection process. The simulation that is
imposed to validate if the noise distribution meets the guideline criteria is
created with a database size of eleven Chemicals yielding $2^{Chem}-1=2047$
chemical combinations , to process $6000$ randomly picked chemicals. The
correlated noise algorithm applied the following noise equations 28 ,28, and
28. A normalization for element intensity in equation 33 was imposed, since
the absorbance can vary so greatly, $N_{EI}$.

$\textbf{N}_{EI}=\frac{\textbf{S}}{\textsl{max}(\textbf{S})}$ (33)

In order to create a robust synthetic noise model, we need to account for
dilation and compression equally. We are able to note the displacement of the
intensity to the corresponding elements with a histogram representation of
$\Delta\textbf{I}_{E}$, defined as 34. The parameter $\eta$ is varied over
four simulation to evaluate the effects of the variance on the distribution,
as shown in Figure 9, identifying the asymmetry in the noise distribution.
This current noise model does not meet the guidelines stated and is therefore
non-ideal. The model provided would provide questionable results in the
detection algorithm, where it is ambiguous if the algorithm failed or if it
was a slight change in $\eta$ causes a large shift in the attenuation of the
original signal.

$\Delta\textbf{I}_{E}=\textbf{J}_{\hat{CD}}\hskip
2.84526pt\textit{diag}(\textbf{N}_{EI})-\textbf{N}_{EI}$ (34)

Figure 9: Histogram of the Change in Intensity of Corresponding Elements.

Figure 10 examines the change in intensity over the entire vector, which
solidifies the need for the stated guidelines of $(4)$ and $(5)$. We can can
see that from the distribution that a shifting is occurring as $\eta$
increases, preventing any type of dilation to occur within the spectrum. The
change of intensity over the entire vector, $\textbf{N}_{VI}$ , of the
normalized spectrum, $\textbf{N}_{VI}$, was calculated by equations 35 and 36,
to achieve the histogram Figure 10.

$\displaystyle\textbf{N}_{VI}=\frac{\textbf{S}}{\sum_{k=1}^{N}(\textbf{S})}$
(35)
$\displaystyle\Delta\textbf{I}_{V}=\sum_{k=1}^{N}(\textbf{J}_{\hat{CD}}\hskip
2.84526pt\textit{diag}(\textbf{N}_{VI})-\textbf{N}_{VI})$ (36)

Figure 10: Histogram of the Change in Intensity of Entire Vector.

The cause can be further explained by examining the expectation of the
designed noise vector $W_{i,1}$ elements’ $j$ and the adjacent windowing
vectors.

$\displaystyle E[\hat{\textbf{W}}_{{i,1}_{j}}]$ $\displaystyle=E[\eta\cdot
X\cdot W_{{i,1}_{j}}+U_{j}-\frac{\eta}{2}U_{j}]$ $\displaystyle
E[\hat{\textbf{W}}_{{i,1}_{j}}]$ $\displaystyle=\eta\cdot
W_{{i,1}_{j}}E[X]+E[U_{j}]-E[\frac{\eta}{2}U_{j}]$ $\displaystyle
E[\hat{\textbf{W}}_{{i,1}_{j}}]$ $\displaystyle=\eta\cdot
W_{{i,1}_{j}}E[X]+E[1]-E[\frac{\eta}{2}]$ $\displaystyle
E[\hat{\textbf{W}}_{{i,1}_{j}}]$ $\displaystyle=\eta\cdot
W_{{i,1}_{j}}\frac{1}{2}+1-\frac{\eta}{2}$

The adjacent windowing elements are constants and do not need to be evaluated
($W_{i,3}$ $W_{i,4}$), but should be examined pictorially with $W_{i,1}$ in
Figure 11.

Figure 11: Windowing Segment Examining Expectation.

We can now note, pictorially from Figure 11, that none of the elements
expectations surpass one, ultimately causing a summed loss of intensity or
compression of the signal. We can also note that anytime these adjacent
windows occur ($W_{i,2}$ $W_{i,3}$ $W_{i,4}$), we will always encounter a
further induced intensity loss. This intensity lose is not equally compensated
by dilation. Furthermore, even when dilation occurs within the function 28 it
still compresses the width of the peak, even though it’s an effective method
for dilation. This deficiency can be attributed to the innate way we window,
since the peak of the signal can range anywhere with the window length
$L_{w}$. However, we should not see this as a shortcoming since it causes
further realist noise by shifting the spectrum by a function of $\eta$.

###  4.5 Approximating an Advantageous Synthetic Noise

We propose the following combined dilation and compression noise functions to
fulfill the following guidelines, based on the windowing foundation function
20. In order to satisfy these requirements, constraints are imposed to the
functions. However, we maintain the functions robustness to dynamically be
altered for compression and dilation.

Figure 12: Simulated Optimized Noise.

(a) Element Intensity

(b) Vector Intensity

Figure 13: Histograms of Intensity.

####  4.5.1 Defining a Robust function

This is done by using a binomial distribution to create an indictor function I
to determine what $\textbf{W}_{i,1}$ function to implement, compression
$\hat{\textbf{W}}_{C_{i,1}}$ or dilation $\hat{\textbf{W}}_{D_{i,1}}$, where
$p=.5$ , in equation 38. This allows further manipulation of the function for
us to engage the guidelines criteria more stringently by enabling two
functions to competent against each other to achieve an approximate zero mean
distribution.

$\textmd{I}=\left\\{\begin{array}[]{ll}1,&\mbox{with probablity p}\\\
0,&\mbox{with probablity (1-p)}\\\ \end{array}\right.$ (37)

The adjacent window’s elements are dependent on the design of
$\hat{\textbf{W}_{i,1}}$. We designed $\hat{\textbf{W}_{i,1}}$ from the
fundamentals of 21 and 25 for the compression and dilation vectors. Thus, the
following equations 39, 40 were developed for the $\textbf{W}_{i,1}$ windowing
case.

$\displaystyle\hat{\textbf{W}}_{B_{i,1}}=\hat{\textbf{W}}_{C_{i,1}}\textmd{I}+\hat{\textbf{W}}_{D_{i,1}}(\textmd{I}-1)$
(38)
$\displaystyle\hat{\textbf{W}}_{C_{i,1}}=(A_{c1}X_{2}\eta)\textbf{W}_{i,1}+\textbf{U}_{L_{w}}(1-A_{c2}\eta+A_{c3}\eta
X_{3})$ (39)
$\displaystyle\hat{\textbf{W}}_{D_{i,1}}=(A_{d1}X_{1}\eta)\textbf{W}_{i,1}+\textbf{U}_{L_{w}}$
(40)

where $\eta\in[0,1],X_{i}$ is a uniform random variable $\in[0,1],\forall
i\in\overline{1,3}$, and $A_{x}i$ are coefficients to control the functions.
In order to avoid discontinuities between the adjacent windows, due to biasing
from each window from the random variable $X_{3}$, the following adjustments
were made to the following cases:

$\displaystyle\hat{\textbf{W}}_{i,2}=\textbf{W}_{i,2}(1-\hat{\textbf{W}}_{i+1,1}(1))+\hat{\textbf{W}}_{i+1,1}(1),\hskip
22.76219pt\mbox{for $(\gamma)(\beta)\leqslant j\leqslant(\gamma)$}$ (41)
$\displaystyle\hat{\textbf{W}}_{i,2}=\textbf{W}_{i,2}(1-\hat{\textbf{W}}_{i-1,1}(L_{W}))+\hat{\textbf{W}}_{i-1,1}(L_{W}),\hskip
22.76219pt\mbox{for $1\leqslant j\leqslant\frac{\alpha(\gamma)}{2}$}$ (42)
$\displaystyle\hat{\textbf{W}}_{i,3}=\textbf{W}_{i,3}(1-\hat{\textbf{W}}_{i+1,1}(1))+\hat{\textbf{W}}_{i+1,1}(1),$
(43)
$\displaystyle\hat{\textbf{W}}_{i,4}=\textbf{W}_{i,4}(1-\hat{\textbf{W}}_{i-1,1}(L_{W}))+\hat{\textbf{W}}_{i-1,1}(L_{W}),$
(44)

####  4.5.2 Constraint A:

Constraint A is to bound the expected dilation and compression functions to be
equivalent. This will assist one functions maximum from overpowering the other
and maintain a equivalent magnitude changes.

$\displaystyle 1-E[\min_{j\in\overline{1,L_{W}}}(\hat{W}_{C_{i,1}j})]$
$\displaystyle=E[\max_{j\in\overline{1,L_{W}}}(\hat{W}_{D_{i,1}})]-1$
$\displaystyle 1-E[1-A_{c2}\eta+A_{c3}\eta X_{3}]$
$\displaystyle=E[A_{d1}X_{1}\eta+1]-1$ $\displaystyle
1-E[1]+E[A_{c2}\eta]-E[A_{c3}\eta X_{3}]$
$\displaystyle=E[A_{d1}X_{1}\eta]+E[1]-1$ $\displaystyle
E[A_{c2}\eta]-E[A_{c3}\eta X_{3}]$ $\displaystyle=E[A_{d1}X_{1}\eta]$
$\displaystyle\eta A_{c2}-A_{c3}\eta E[X_{3}]$ $\displaystyle=A_{d1}\eta
E[X_{1}]$ $\displaystyle A_{c2}-A_{c3}E[X_{3}]$ $\displaystyle=A_{d1}E[X_{1}]$
$\displaystyle A_{c2}-\frac{1}{2}A_{c3}$ $\displaystyle=\frac{1}{2}A_{d1}$
$\displaystyle 2A_{c2}-A_{c3}$ $\displaystyle=A_{d1}$

####  4.5.3 Constraint B:

Constraint B is to bound the variance of the dilation and compression
functions to be equivalent. This will prevent the dilation and compression
functions from causing an asymmetry within the noise distribution.

$\displaystyle Var[\min_{j\in\overline{1,L_{W}}}(\hat{W}_{C_{i,1}j})]$
$\displaystyle=Var[\max_{j\in\overline{1,L_{W}}}(\hat{W}_{D_{i,1}})]$
$\displaystyle Var[1-A_{c2}\eta+A_{c3}\eta X_{3}]$
$\displaystyle=Var[A_{d1}X_{1}\eta+1]$ $\displaystyle Var[A_{c3}\eta X_{3}]$
$\displaystyle=Var[A_{d1}X_{1}]$ $\displaystyle A_{c3}^{2}\eta^{2}Var[X_{3}]$
$\displaystyle=A_{d1}^{2}\eta^{2}Var[X_{1}]$ $\displaystyle A_{c3}^{2}$
$\displaystyle=A_{d1}^{2}$

####  4.5.4 Constraint C:

Constraint C is to bound the compression function’s maximum element in that
vector to never surpass the maximum value of the dilation function’s maximum
element.

$\displaystyle\min_{j\in\overline{1,L_{W}}}(\hat{W}_{C_{i,1}j})$
$\displaystyle\leq\max_{j\in\overline{1,L_{W}}}(\hat{W}_{D_{i,1}})$
$\displaystyle A_{c1}\eta+1-A_{c2}\eta+A_{c3}\eta X_{3}$ $\displaystyle\leq
A_{d1}X_{1}\eta+1$ $\displaystyle A_{c1}\eta-A_{c2}\eta+A_{c3}\eta X_{3}$
$\displaystyle\leq A_{d1}X_{1}\eta$ $\displaystyle A_{c1}-A_{c2}+A_{c3}$
$\displaystyle\leq A_{d1}$

####  4.5.5 Constraint D:

Recalling Figure 11, based on the topology of design there will always be a
greater intensity loss with the compression function is implemented. To
achieve an approximate zero mean intensity loss for the vector, stated by the
fifth guideline, we minimize the expected intensity loss of the vector when
compared in a random process of dilation and compression by exploiting the
binomial equation. Using the binomial equation to our advantage, we can design
the $\hat{\textbf{W}}_{C_{i,1}}$ and $\hat{\textbf{W}}_{D_{i,1}}$ functions
accordingly having them achieve an approximate averaged expected intensity
value over all elements, 45. In order to account for this average expectation,
we account for the effects of the adjacent windows as well for a proper
approximation. In the dilations case we used
$\textbf{W}_{D}=[\textbf{W}_{i,5},\hat{\textbf{W}}_{i,1},\textbf{W}_{i,5}]$
and for the compression case we used
$\textbf{W}_{C}=[\hat{\textbf{W}}_{i,3},\hat{\textbf{W}}_{i,1},\hat{\textbf{W}}_{i,4}]$.
Therefore, base on the design of $\hat{\textbf{W}}_{i,1}$ for the compression
and dilation we can manipulate our indictors function’s $p$ value to further
optimize the expected intensity to have an approximate equivalent intensity
deviation of compression and dilation.

$E_{win}=p\frac{1}{3*L_{w}}(\sum_{j=1}^{3\cdot
Lw}E[W_{C}])+(1-p)\frac{1}{3*L_{w}}(\sum_{j=1}^{3\cdot Lw}E[W_{D}])\approx 1$
(45)

The following approximation was done with
$\alpha=1,\eta=.1,A_{c1}=A_{c2}=A_{c3}=A_{d1}=.5$ and $L_{w}=100$, where
$E_{win}=0.9918$ with a $p=.33$, implementing this optimize noise at these
parameters we manifest others noise vectors at various different $\eta$’s
assuming the changes are minute. If desired you can optimize the $p$ for
different $eta$s. In Figure 12, we used the aforementioned parameters to
designed noise at $\eta=1$, where simulations of the distributions of element
and vector intensity is shown in 13.

##  5 Conclusion

The synthetic noise generation was designed to create to mimic spectrum’s
changes caused by user error, stray light, wavelength accuracy, and self
absorption. As you can note, that additive white Gaussian noise simply can not
use to mimic such erroneous changes within the spectroscopy signal. The noise
that was created is correlated to the various peaks within the spectrum and
then constrained by the percentage of error by the parameter $\nu$. More
completed work that highlights the full creation of these spectroscopy data
sets and probably detection applications are available [21, 22].

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