# MACHINE LEARNING APPLIED TO PERUVIAN VEGETABLES IMPORTS

Ticona-Salluca Hugo  Faculty of Statistic and Computer Engineering,
Universidad Nacional del Altiplano de Puno, P.O. Box 291
Puno - Peru.
Email<EMAIL_ADDRESS>Torres-Cruz Fred  Faculty of Statistic and
Computer Engineering,
Universidad Nacional del Altiplano de Puno, P.O. Box 291
Puno - Peru.
Email<EMAIL_ADDRESS>Tumi-Figueroa Ernesto Nayer  Faculty of Statistic
and Computer Engineering,
Universidad Nacional del Altiplano de Puno, P.O. Box 291
Puno - Peru.
Email<EMAIL_ADDRESS>
###### Abstract

The current research work is being developed as a training and evaluation
object. the performance of a predictive model to apply it to the imports of
vegetable products into Peru using artificial intelligence algorithms,
specifying for this study the Machine Learning models: LSTM and PROPHET. The
forecast is made with data from the monthly record of imports of vegetable
products(in kilograms) from Peru, collected from the years 2021 to 2022. As
part of applying the training methodology for automatic learning algorithms,
the exploration and construction of an appropriate dataset according to the
parameters of a Time Series. Subsequently, the model with better performance
will be selected, evaluating the precision of the predicted values so that
they account for sufficient reliability to consider it a useful resource in
the forecast of imports in Peru.

###### Index Terms:

Machine Learning: forecasting; time series; imports;artificial
intelligence;dataset

##  I Introduction

The Machine Learning methodology is one of the most valuable technologies in
the Industry 4.0, and advances in Machine Learning have provided significant
benefits in strategic decision-making in organizations[1]. In recent years,
predictive models using Machine Learning algorithms have been implemented in
real environments of different organizations, and as in this particular case
that concerns us, it also operates in the agriculture industry and optimal
results are expected from these technologies.

Likewise, Machine Learning technology has a high potential to optimize your
processes and, consequently, make correct and anticipated decisions. For this
reason, Learning Algorithms should also be considered as the next innovative
direction to significantly improve the prediction of these use cases. [2].

Therefore, this research will seek to demonstrate and encourage the use of
Machine Learning applications in the field of agribusiness in Peru. Through
the present work, the examination and modeling of real data will be developed
in order to forecast future values with the least possible prediction error.
In this sense, this applied research work covers the design, training,
validation, and evaluation of 2 predictive models. The research models to be
explored are LSTM and Prophet models, these being adequate and recommended to
develop applied foresting, on the one hand, LSTM is recognized for its
hyperparameters which have the ability to capture non-linear patterns using
the grid search method[3] and PROPHET for their modular regression with
intuitive and adjustable interpretation parameters applied on Time Series[4].

.

##  II DATASET

According to the purpose of the study, the official records of the export
volume (in kilograms) of vegetables in Peru were used, collected from 2021 to
2022 in the National Open Data Platform,[5], officially uploaded by the
Ministry of Agrarian Development and Irrigation of Peru.

Table 1
Description of the variables in the Vegetable Import Dataset in Peru
2021-2022.

Based on this dataset, it is proposed to make a reliable prediction, and
according to the models to be developed, we will therefore need the continuous
quantitative variable, PESO.

Also according to the observed data, we find it necessary to adapt a proper
timeline; therefore, we must join the variables AÑO and MES to foresee a more
accurate seasonality, specifying the variables to forecast. Subsequently, to
get a more accurate overview of our data regarding products, we define IDs for
each product.

Figure 1: Percentage description of each vegetable product imported in Peru

From the dataset, we highlight 3P: Soja, cake with 23.1% in 1st position, in
second position 5P: Apple, fresh fruit with 6.3% and in third position 27P:
Soja, grain with 4.2%. The data ranges from 2021-05-01 to 2022-06-01, with
34,364 observations on imports of a total of 848 plant products.

##  III METHODS

In this section, we will describe the models used for the predictions.

###  III-A LSTM MODEL

The LSTM name refers to the analogy that a standard RNN has both ’long-term
memory’ and ’short-term memory’. The connection weights and biases in the
network change once per training episode, analogous to how physiological
changes in synaptic strengths store long-term memories; activation patterns in
the network change once per time step, analogous to how moment-to-moment
changes in electrical activation patterns in the general brain store short-
term memories. The LSTM architecture aims to provide a short-term memory for
RNNs that can last for thousands of time steps and continue to be reliable,
hence ’short-term long-term memory’[6].

LSTM is considered a special type of recurrent neural network (RNN), developed
to solve the potential problem of descending gradient found in traditional RNN
training, and is able to learn both short-term and long-term dependencies[7]
and is constructed of four main components: an entry gate, an exit gate,
memory cell and a forget gate.

Input Gate: controls the sending addition of information to the cell state. In
other words, the gateway will consider what information needs to be added to
the cell state to ensure that important information is added and that there is
no redundant information or noise.

Memory Cell: controls the value that might be deleted or updated, and contains
a value that might need to be kept as additional information for many other
time steps.

Output Gate: controls the selection of useful learning information from the
current state of the cell as output.

Forget Gate: controls the removal of information that is no longer needed for
LTSM to learn things or, less importantly, the state of the cell. This helps
to optimize the performance of the LSTM proposed network.

Also, The LSTM model follows the sequence:

1\. Decide to discard the cell state information by a Sigmoid layer called
”forget gate”.

2\. Decide new information to store in the cell state. The Sigmoid layer
called the ”gateway layer” decides which values will be updated. the tan coat
creates a vector of new candidate values that could be added to the state.

3\. Update the old cell state to the new cell state.

4\. Decide the filtered output based on the state of the cell.

Figure 2: MODEL LSTM

Considered for a normal LSTM model is the Tanh property, which is a non-linear
activation function. It regulates the values that flow through the network,
keeping the values between -1 and 1. To avoid information fading, a function
is needed whose second derivative can survive longer. There might be a case
where some values become huge, causing the values to be irrelevant.

And of course an LSTM principal property: The Sigmoid that belongs to the
family of nonlinear activation functions. Unlike Tanh, the Sigmoid holds
values between 0 and 1. It helps the network update or forget data. If the
multiplication results in 0, the information is considered forgotten.
Similarly, the information is kept if the value is relevant[8].

This will assist the network in determining what data can be lost and what
data should be kept.

###  III-B PROPHET MODEL

PROPHET is a procedure for forecasting time series data based on an additive
model in which nonlinear trends are adjusted for annual, weekly, and daily
seasonality. It works best with time series that have strong seasonal effects
and multiple seasons of historical data.

In the specification of this Prophet model, there are several places where we
can alter the model to apply your experience and external expert knowledge
without needing to understand the underlying statistics.[4]

For the Prophet model with general theory:
We use a mathematical decomposable time series model[9] and this model has
these components: trend, seasonality, and holidays. They are combined into the
following equation:

Series Model Formula.

Here g(t) is the trend function that models non-periodic changes in the value
of the time series, s(t) represents periodic changes (for example, weekly and
yearly seasonality), and h(t) represents the effects of the seasons, The error
term represents any idiosyncratic changes that do not fit the model.

In PROPHET we incorporate trend changes into the growth model by explicitly
determining the change points where the growth rate is allowed to change.
Suppose there are exchange points at moments j,j= 1,…, S. We give N a vector
of adjustments, where only the rate change occurs at moment j. The rate at any
time is then the base rate k, plus all adjustments up to the point of This is
best represented by the vectors as follows:

Adjustments represented.

When rate k is adjusted, the parameter set must also be adjusted to connect
the endpoints of the segments. The correct fit at the shift point is easily
calculated as:

That would be the Adjust at shift point.

The logistic piece of the growth model then looks as follows:

The Fourier series is also applied in Prophet to provide a flexible model of
periodic effects[10]. Let P be the regular period that we expect the time
series to have (for example, P= 365:25 for annual data or P= 7 for weekly
data, when we scale our indices of time variables). We can approximate
arbitrary uniform seasonal effects with this definition:

Fourier Definition

Sometimes we can’t just randomly split the data accordingly. PROPHET develops
simulated historical forecasts by producing K forecasts at various cut-off
points in history, chosen such that the horizons are within the historical
record and the total error can be evaluated.

This procedure is based on the classic ’rolling source’ forecast evaluation
procedures[11], but uses only a small sequence of target dates instead of
forecasting by historical date) is that it saves on computation and provides
less correlated precision measurements.

###  III-C Work Sequence

  * •

We apply an EDA (Exploratory Data Analysis) it is necessary to clean the data
and adapt it for the job.

  * •

Normalization of the data, we adapt the data to follow a supervised sequence
model according to a time series.

  * •

Let’s divide our dataset into proof and validation tests.

  * •

Coding and implementation of the models according to the Prophet or LSTM case.

  * •

Adjustment of the parameters and extraction of predictions with their
respective evaluation metrics.

##  IV RESULTS

The results using PROPHET and LSTM are shown in this section together with the
general comparison of the mentioned models.

We will also appreciate the comparisons and metrics developed according to the
models applied to the study variable: PESO.

Result according to the validation of the LSTM model:

Figure 3: LSTM VALIDATION

Result according to the validation of the PROPHET model:

Figure 4: PROPHET VALIDATION

Comparison to the validation of the PROPHET and LSTM model:

Figure 5: LSTM AND PROPHET COMPARISON

MSE(mean squared error) and the RMSE(root mean squared error) of both models:

We can appreciate each model according to its evaluation.

Figure 6: RMSE Y MSE

##  V Discussion and Conclusions

In keeping with the theory, all machine learning algorithms are unique, which
is the root cause of why the prediction results are different algorithms on
the same data set differ. LSTM was proposed for being an improvement of the
RNNs and Prophet for its versatility in data with less presence of
seasonality.
Given what was taken into consideration for LSTM, the applied model was with
the minimum parameters and show results with much better conditioning than was
expected, remembering that in different studies the superiority of LSTM is
detailed over the basic algorithms that apply Recurrent Neural Networks.
Furthermore, it is noted from the theory that the number of training times,
known as the ’epoch’ in learning[12], did not take into account the expected
effect on the performance of the trained forecast model and exhibited mostly
random behavior. How intuitive The developed model based on LSTM incorporates
additional ’gates’ in order to store longer sequences of input data.
One of the main questions when developing and analyzing is whether the gates
incorporated in the LSTM architecture would give a good prediction and if
additional data training would be needed to further improve the prediction[13]
and in this case, we can deduce that the quantity of data was acceptable but
the predictions were affected by the not very well defined seasonality of the
dataset. A more effective solution would be to add exact dates and continuous
seasonalities.

For the Prophet model, we should have some more intuitive results according to
its theory, the application of the Fourier series could develop more
precision. The model is expected to obtain a reasonable forecast on disordered
data without too much manual effort, unlike LSTM, which has more
hyperparameters. Prophet proved to be resistant to outliers, missing data, and
drastic changes in its time series, the intention to fit the timeline is
noted. Compared to other classical forecasting methods, Prophet should be fast
and easy to apply to time series, which is what it was designed for in the
first place; however, it could be considerably less accurate[14] and in this
case, we confirm this appreciation by highlighting once again its intuitive
factor.

The Prophet procedure should include more parameters for users to modify and
adjust the forecasts in a more effective way. Also, a hybrid model could
improve significantly.

According to the RMSE results of the import predictions, we can conclude that
the LSTM model presents a significantly better performance and reliability
with respect to the Prophet model, however, as we deduced previously, the
seasonality of the dataset was an important key in the variation of the
development of the models and their predictions. Therefore, increasing the
size of the dataset and adapting an exact timeline for our dataset of
vegetable imports from Peru should be a priority, in this way, we would
undoubtedly obtain results with better relevance and reliability and, of
course, the field of application with the use of machine learning techniques
would be widely used and its results would be of a strongly necessary
relevance.

## References

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# A simple model for citation curve

Y.C. Tay

Mostafa Rezazad

Hamid Sarbazi-Azad

Dept. of Computer Science  National Univ. of Singapore  Singapore  School of
Computer Science  Institute for Research in Fundamental Sciences (IPM)
Tehran, Iran  Sharif Univ. of Technology and  Institute for Research in
Fundamental Sciences (IPM)  Tehran, Iran

###### Abstract

There is considerable interest in the citation count for an author’s
publications. This has led to many proposals for citation indices for
characterizing citation distributions. However, there is so far no tractable
model to facilitate the analysis of these distributions and the design of
these indices. This paper presents a simple equation for such design and
analysis. The equation has three parameters that are calibrated by three
geometrical characteristics of a citation distribution. Its simple form makes
it tractable. To demonstrate, the equation is used to derive closed-form
expressions for various citation indices, analyze the effect of time and
identify individual contribution to the Hirsch index for a group.

##  1 Introduction

Since the launch in 2004 of the web search engine Google
Scholar111https://scholar.google.com/, it has become easy to look for the
papers and publication record of a researcher. The information provided
currently includes the citations for each publication, and a $h$-index for the
citation count.

The $h$-index was proposed by Hirsch in 2005 [9]. It gained much attention and
triggered numerous proposals for alternative citation indices [4], but there
is controversy over characterizing an author’s research record by such indices
[2].

In this paper, we do not advocate one index or another, and propose none
ourselves. Instead, we offer a simple equation for approximating the
distribution of citation count. We claim that this equation can facilitate the
analysis of these distributions, and the design of citation indices.

To demonstrate our claim, we use the equation to derive closed-form
expressions for various indices in Sec. 3. In Sec. 4, we further apply the
equation to examine the effect of time, and how individual $h$-indices
contribute to the $h$-index of a group of researchers.

##  2 The proposed model

Let $\Psi(n)$ denote the number of citations for an author’s $n$-th
publication, where the publications are sorted based on their citation numbers
so that $\Psi(n)\geq\Psi(n^{\prime})$ for $n<n^{\prime}$. Let $M=\Psi(1)$,
i.e. the maximum number of citations for the author’s most cited publication.
Suppose the author has $N$ cited publications, so $\Psi(N)>0$ but $\Psi(N+1)$
is either 0 or undefined. We seek a closed-form expression to define a
function $f$ that approximates $\Psi$.

A frequently-used expression [8] is the power law

$f(x)=\frac{C}{x^{\lambda}},$ (1)

where $C$ and $\lambda$ are parameters that vary among authors, $C>0$ and
$\lambda>0$. This $f:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ has 3 issues:

First, the vertical asymptote at $x=0$ can make the approximation bad for
authors who do not have hugely different citation counts for top-ranked
papers. Second, the horizontal asymptote as $x\rightarrow\infty$ can give a
poor approximation for authors with a small number of papers $N$. Third, two
parameters do not suffice: We can think of $M$ and $N$ as anchoring $f$, but
they leave much ambiguity for the curvature in between.

(For the power law, if we require $f(1)=M$, then $C=M$ in Eqn. (1), and
curvature is determined by $\lambda$.)

We therefore need at least 3 parameters to specify $f$. We use the point where
$f$ cuts the diagonal line to fix the curvature for $f$, i.e. if the real
value $h$ is defined by $f(h)=h$, then $f$ is determined by $M$, $N$ and $h$.

What should we choose for $f$? For the power law, $1/f(x)$ is proportional to
$x^{\lambda}$. Fig. 1 plots $1/\Psi(n)$ for 5 researchers in engineering
($M=718,N=171,h=50$), mathematics ($M=8763,N=410,h=77$), medicine
($M=2019,N=345,h=114$), psychology ($M=3740,N=982,h=243$) and sociology
($M=1057,N=116,h=24$). The citation data is from a dataset of 226 authors that
we sampled from Publish or Perish222https://harzing.com/resources/publish-or-
perish; the histograms for $M$, $N$ and $h$ are in the Appendix. We use this
dataset throughout this paper.

Figure 1:  Regression lines show that $1/\Psi(n)$ is approximately linear for
small $n$. (The research area for each author is indicated. Note the nonzero
intercepts.)

In each case, the regression line shows that $1/\Psi(n)$ is approximately
linear in $n$ for small $n$, where most of an author’s citations are. This
suggests $f$ should have the form $1/f(x)=\gamma_{1}x+\gamma_{0}$ for some
constants $\gamma_{0}$ and $\gamma_{1}$. Since $f$ needs to have 3 parameters,
we define it as

$f(x)=\frac{b}{x+c}-a.$ (2)

where $a$, $b$ and $c$ are positive real values.

Figure 2:  The shape for $f(x)$ is determined by $M$, $N$ and $h$, which
calibrate $a$ for the horizontal asymptote, $c$ for the vertical asymptote,
and $b$ for the curvature.

Fig. 2 illustrates this $f$. The function has a horizontal asymptote at
$y=-a$, vertical asymptote at $x=-c$, and intersects $y=x$ at $x=h$. These 3
parameters ($a,b,c$) control the location and curvature of $f$. Their values
are determined by

$f(0)=M,\quad f(N)=0\quad{\rm and}\quad f(h)=h.$ (3)

It follows that $f(1)\approx f(0)=M=\Psi(1)$, and $f(N)=0\approx\Psi(N)$.
Moreover, the $h$-index [9] is defined by solving $\Psi(n)=n$, so
$f(h)=h\approx\Psi(h)$. We are only interested in $f(x)$ for $0\leq x\leq N$.

Solving Eqn. (2) and Eqn. (3) gives

$\displaystyle\begin{split}a&=\frac{Mh^{2}}{MN-(M+N)h}\\\
b&=MN(M-h)(N-h)\big{(}\frac{h}{MN-(M+N)h}\big{)}^{2}\\\
c&=\frac{Nh^{2}}{MN-(M+N)h}\\\ \end{split}$ (4)

Fig. 3 shows how well $f(x)$ fits $\Psi(n)$ for 6 researchers from our
dataset.

(i) small $M$ (ii) large $M$

(iii) small $N$ (iv) large $N$

(v) small $h$ (vi) large $h$

Figure 3:  Comparing number of citations $\Psi(n)$ and approximation $f(x)$
for 6 different authors whose research areas are in computer science,
chemistry and medicine.

###  2.1 Approximations for the head and tail

Let ${\Psi_{\rm head}}$ and ${\Psi_{\rm tail}}$ denote the ${\Psi}$ for $n\leq
h$ and $n>h$ respectively. One can simplify the expressions in Eqn. (4) by
focusing on ${\Psi_{\rm head}}$ and neglecting the fit for ${\Psi_{\rm
tail}}$. We do this by taking the limit $N\rightarrow\infty$ in Eqn. (4) and
thus derive another approximation ${f_{\rm head}}$ from $f$:

$\displaystyle\begin{split}{a_{\rm head}}&=0\\\ {b_{\rm
head}}&=\frac{Mh^{2}}{M-h}\\\ {c_{\rm head}}&=\frac{h^{2}}{M-h}\\\ {\rm
and}\quad{f_{\rm head}}(x)&=\frac{{b_{\rm head}}}{x+{c_{\rm
head}}}.\end{split}$ (5)

In effect, ${f_{\rm head}}$ is obtained from ${\Psi}$ by solving

${f_{\rm head}}(0)=M,\quad{f_{\rm head}}(h)=h\quad{\rm
and}\quad\lim_{x\to\infty}{f_{\rm head}}(x)=0$

using $M$ and $h$ to calibrate the two parameters ${b_{\rm head}}$ and
${c_{\rm head}}$ for ${f_{\rm head}}$. Fig. 4 shows how ${f_{\rm head}}$ can
give a better fit if $f$ over-estimates $\Psi$.

Figure 4:  ${f_{\rm head}}$ can provide a better fit for $\Psi(n)$,
$n=1,\ldots,h$. (The author is a computational biologist.)

We can similarly extract an approximation ${f_{\rm tail}}$ for ${\Psi_{\rm
tail}}$ by taking $M\rightarrow\infty$. This gives

$\displaystyle\begin{split}{a_{\rm tail}}&=\frac{h^{2}}{N-h}\\\ {b_{\rm
tail}}&=\frac{Nh^{2}}{N-h}\\\ {c_{\rm tail}}&=0\\\ {\rm and}\quad{f_{\rm
tail}}(x)&=\frac{{b_{\rm tail}}}{x}-{a_{\rm tail}}.\end{split}$ (6)

Equivalently, we use $N$ and $h$ to calibrate parameters ${a_{\rm tail}}$ and
${b_{\rm tail}}$ in ${f_{\rm tail}}$ by solving

${f_{\rm tail}}(N)=0,\quad{f_{\rm tail}}(h)=h\quad{\rm and}\quad\lim_{x\to
0}{f_{\rm tail}}(x)=\infty.$

###  2.2 Areas under the curve

Let ${\Theta}$ denote the total number of citations for an author’s
publications. Using $f$, we can approximate ${\Theta}$ by

$\displaystyle F$
$\displaystyle=\int_{0}^{N}f(x)dx=\int_{0}^{N}\big{(}\frac{b}{x+c}-a\big{)}dx$
$\displaystyle=b\ln{\big{(}1+\frac{N}{c}\big{)}}-aN$
$\displaystyle=\frac{MN(M-h)(N-h)h^{2}}{(MN-(M+N)h)^{2}}\ln{\big{(}1+\frac{MN-(M+N)h}{h^{2}}\big{)}}$
$\displaystyle\quad-\frac{MNh^{2}}{MN-(M+N)h}$ (7) $\displaystyle\approx
h^{2}\ln{\big{(}\frac{MN}{{\tt e}h^{2}}\big{)}}\quad{\rm for}\ M>>h\ {\rm
and}\ N>>h.$ (8)

Here, we use ${\tt e}$ to denote Euler’s number (${\tt e}\approx 2.71828$), to
avoid confusion with $e$ in the $e$-index below.

(a) Eqn. (7) over-estimates $F$ (on average) (b) Eqn. (8) under-estimates $F$
(on average)

Figure 5:  Total number of citations: comparing approximate value $F$ to
empirical value ${\Theta}$. (In this and the following plots, the regression
lines are constrained to pass through 0.)

The above approximations for $F$ are plotted against ${\Theta}$ in Fig. 5 for
our dataset of 226 authors; if $F$ is an accurate estimate for ${\Theta}$,
then the sample points would scatter around the diagonal line $F={\Theta}$.
Fig. 5 shows that Eqn. (7) over-estimates ${\Theta}$ (on average), since the
regression line has a gradient of 1.15, whereas Eqn. (8) under-estimates
${\Theta}$ (on average), since the regression line has a gradient of 0.88.

In Fig. 5(a), there is an obvious outlier at
$({\Theta},F)=(39868,4943)$; it is from a computer scientist with an extremely
skewed $\Psi$ that drops from $\Psi(1)=28142$ to $\Psi(10)=102$. This author
appears as outlier in most of the following plots as well.

One could improve on the approximation by shifting $f$ so $f(1)=M$ (instead of
$f(0)=M$), but that would further complicate the expressions for $a,b$ and $c$
in Eqn. (4).

Let ${\Theta_{\rm head}}$ denote the total number of citations for an author’s
first $h$ publications. We can use ${f_{\rm head}}$ to approximate
${\Theta_{\rm head}}$ by

$\displaystyle{F_{\rm head}}$ $\displaystyle=\int_{0}^{h}\frac{{b_{\rm
head}}}{x+{c_{\rm head}}}dx={b_{\rm head}}\ln{\big{(}1+\frac{h}{{c_{\rm
head}}}\big{)}}$
$\displaystyle=\frac{Mh^{2}}{M-h}\ln{\big{(}1+\frac{M-h}{h}\big{)}}$
$\displaystyle=\frac{Mh^{2}}{M-h}\ln{\big{(}\frac{M}{h}\big{)}}$
$\displaystyle=h\left(\frac{\ln{\frac{1}{h}}-\ln{\frac{1}{M}}}{\frac{1}{h}-\frac{1}{M}}\right)$
(9)

Similarly, if ${\Theta_{\rm tail}}$ denotes the total number of citations for
the $N-h$ publications in the tail, then we can use ${f_{\rm tail}}$ to
approximate ${\Theta_{\rm tail}}$ by

$\displaystyle{F_{\rm tail}}$ $\displaystyle=\int_{h}^{N}\big{(}\frac{{b_{\rm
tail}}}{x}-{a_{\rm tail}}\big{)}dx={b_{\rm
tail}}\left(\ln{\frac{N}{h}}\right)-{a_{\rm tail}}(N-h)$
$\displaystyle=\frac{Nh^{2}}{N-h}\left(\ln{\frac{N}{h}}\right)-h^{2}.$ (10)

##  3 Closed-form expressions for indices

In this section, we relate $f$, ${f_{\rm head}}$ and ${f_{\rm tail}}$ to
previous work. In particular, we use our equations to derive closed-form
expressions for various indices.

###  3.1 Total number of citations

When introducing the $h$-index, Hirsch [10] postulated that the total number
of citations

${\Theta}=\alpha h^{2}$ (11)

for some $\alpha$ that varies among authors and, empirically, $3<\alpha<5$.
Eqn. (11) has the equivalent form

$h=\frac{1}{\sqrt{\alpha}}{\Theta}^{0.5}.$

A regression analysis by van Raan [16] using data for chemistry research in
Dutch universities also shows

$h\ =\ 0.42{\Theta}^{0.45},$

where the exponent 0.45 is close to 0.5. (See also the Yong’s “rule of thumb”
[17].) Using our approximation $F$ for ${\Theta}$, Eqn. (8) shows that, in
fact,

$\alpha\approx\ln{\big{(}\frac{MN}{{\tt e}h^{2}}\big{)}}\ ;$

note that $\alpha$ itself depends on $h$.

###  3.2 $A$-index

To take into account $\Psi(n)-h$ for $n=1,\ldots,h$, Jin et al. [11] defined
an $A$-index

$A={\Theta_{\rm head}}/h,$ (12)

i.e. the average citation count for the first $h$ papers. Using Eqn. (9), we
can approximate this $A$-index as

$A\approx\frac{{F_{\rm
head}}}{h}=\frac{Mh}{M-h}\ln{\Big{(}\frac{M}{h}\Big{)}}.$ (13)

Fig. 6 compares the empirical value of the $A$-index for the 226 authors to
the approximate value computed with Eqn. (13).

The regression line has a gradient of 0.86 and correlation coefficient
$R^{2}\approx 0.72$. This weak accuracy is expected, since the $A$-index is
defined with ${\Theta_{\rm head}}$, where there are huge differences among
authors in the shape of $\Psi$ for their highly-cited papers. The outlier at
$(1705,164)$ is from the computer scientist previously mentioned for Fig. 5.
Another outlier at $(455,50)$ is from a chemist with just $N=15$ publications,
but has $M=2259$.

Figure 6:  Comparing approximate value of $A$-index in Eqn. (13) to empirical
value.

Jin et al. proved that, using the power law model $f(x)=M/x^{\lambda}$, $A/h$
is a constant determined only by the curvature parameter $\lambda$. In
contrast, Eqn. (13) shows that, using our model,

$\frac{A}{h}\approx\frac{M}{M-h}\ln{\Big{(}\frac{M}{h}\Big{)}},$

so $A/h$ also depends on the maximum citation $M$.

###  3.3 $R$-index

As an alternative to the $A$-index, Jin et al. [11] also defined an $R$-index

$R=\sqrt{{\Theta_{\rm head}}}.$

By Eqn. (9), we can approximate this as

$R\approx\sqrt{{F_{\rm
head}}}=h\sqrt{\frac{M}{M-h}\ln{\Big{(}\frac{M}{h}\Big{)}}}\ .$ (14)

Fig. (7) shows that, although there are outliers, Eqn. (14) provides a closed-
form expression that, in general, gives an excellent approximation for the
$R$-index (the regression line has gradient 1.00 and $R^{2}\approx 0.97$).

Figure 7:  Comparing approximate value of $R$-index in Eqn. (14) to empirical
value.

Again, under the power law, $R/h$ is expressible in terms of $\lambda$,
whereas Eqn. (14) shows that the ratio depends on $M$ as well.

###  3.4 $g$-index

An author’s $h$-index remains the same no matter how high $\Psi(n)$ is for
$n=1,2,\ldots,h$. To overcome this issue, Egghe defined a $g$-index [8], which
we approximate as

$g\approx\frac{\int_{0}^{g}f(x)dx}{g}.$ (15)

Egghe has shown that $g>h$. We therefore use ${f_{\rm head}}$ and ${f_{\rm
tail}}$ to further approximate $g$ by

$\displaystyle g^{2}$ $\displaystyle\approx\int_{0}^{h}{f_{\rm
head}}(x)dx+\int_{h}^{g}{f_{\rm tail}}(x)dx$
$\displaystyle=\frac{Mh^{2}}{M-h}\ln\big{(}\frac{M}{h}\big{)}+{b_{\rm
tail}}\ln{\big{(}\frac{g}{h}\big{)}}-{a_{\rm tail}}(g-h)$
$\displaystyle\phantom{XXXXXXXXXX}{\rm by\ Eqn.~{}(\ref{eq:abctail})\ and\
Eqn.~{}(\ref{eq:Fhead})}$
$\displaystyle\approx\frac{Mh^{2}}{M-h}\ln\big{(}\frac{M}{h}\big{)}+\frac{Nh^{2}}{N-h}\ln\big{(}\frac{g}{h}\big{)}\quad{\rm
for\ }{a_{\rm tail}}\approx 0$ $\displaystyle\approx
h^{2}\ln\big{(}\frac{M}{h}\big{)}+h^{2}\ln\big{(}\frac{g}{h}\big{)}\quad{\rm
for\ }M>>h\ {\rm and\ }N>>h.$ (16)

Consider some $\beta>\frac{g}{h}$. Then

$\displaystyle\ln\big{(}\frac{g}{h}\big{)}$
$\displaystyle=\ln\beta+\ln\big{(}1-\big{(}1-\frac{g}{\beta h}\big{)}\big{)}$
$\displaystyle\approx\ln\beta-\big{(}1-\frac{g}{\beta h}\big{)}$
$\displaystyle\phantom{XXXXXXX}{\rm since\ }\ln(1-x)\approx-x\ {\rm for\
}0<x<1.$

The citation data we have seen all show $\frac{g}{h}<4$, so we choose
$\beta=4$. (A larger $\beta$ will increase $1-\frac{g}{\beta h}$ and worsen
the approximation.) Therefore,

$\ln\big{(}\frac{g}{h}\big{)}\approx\ln\big{(}\frac{4}{{\tt
e}}\big{)}+\frac{g}{4h}\ .$

Substituting this into Eqn. (16), we get

$\frac{g^{2}}{h^{2}}\approx\ln\big{(}\frac{M}{h}\big{)}+\ln\big{(}\frac{4}{{\tt
e}}\big{)}+\frac{g}{4h}\ ,$

so

$\Big{(}\frac{g}{h}-\frac{1}{8}\Big{)}^{2}-\frac{1}{64}\approx\ln\big{(}\frac{4M}{{\tt
e}h}\big{)}\ .$

Since $\frac{g}{h}>1$, we can further simplify this as

$\displaystyle\Big{(}\frac{g}{h}\Big{)}^{2}$
$\displaystyle\approx\ln\big{(}\frac{4M}{{\tt e}h}\big{)}$ $\displaystyle{\rm
i.e.}\quad g$ $\displaystyle\approx h\sqrt{\ln\big{(}\frac{4M}{{\tt
e}h}\big{)}}$ (17)

Egghe has proven that, for the power law model $f(x)=M/x^{\lambda}$, $g/h$ is
a constant determined by the curvature parameter $\lambda$ only [8]. In
contrast, Eqn. (17) shows that, for our model, $g/h$ also depends on $M$.

Fig. 8 plots the approximation (17) against actual $g$ values for the
previously chosen 226 authors. The regression line shows that, on average, the
approximation is accurate. The under-estimating outlier at $(151,63)$ is from
the computer science previously mentioned for Fig. 5; the over-estimating
outlier at $(320,573)$ is from an immunologist with unusually large $M$ and
$N$ $(M=12172,N=995,h=281)$, for whom ${f_{\rm head}}$ over-estimates
${\Psi_{\rm head}}$.

Figure 8:  Comparing approximate value $g$ in Eqn. (17) to empirical value
$g$.

###  3.5 $hg$-index

Alonso et al. [3] combined the $g$\- and $h$-indices to get an hg-index
$\sqrt{hg}$. By Eqn. (17),

$\sqrt{hg}\approx h\sqrt[4]{\ln\big{(}\frac{4M}{{\tt e}h}\big{)}},$ (18)

so the hg-index just reduces the square root in Eqn. (17) to a 4th root. Fig.
9 shows the closed-form accurately approximates the empirical value of the
$hg$-index.

Figure 9:  Comparing approximate value of $hg$-index in Eqn. (18) to empirical
value.

###  3.6 $e$-index

Zhang [18] pointed out that the $h$-index holds no information for the head,
and its integer value has coarse granularity. To address these issues, he
proposed the $e$-index, defined by

$e^{2}={\Theta_{\rm head}}-h^{2}.$

Using our ${F_{\rm head}}$ approximation (9), we have

$e^{2}\approx\frac{Mh^{2}}{M-h}\ln{\big{(}\frac{M}{h}\big{)}}-h^{2},$ (19)

Fig. 10 shows the value of $e$ computed from Eqn. (19) is a good approximation
of the empirical value for our dataset: The regression line has gradient 1.00
and $R^{2}\approx 0.94$.

Figure 10:  Comparing approximate value of $e$ in Eqn. (19) to empirical value
for the $e$-index.

For the power law model $f(x)=M/x^{\lambda}$, Zhang derived

$e^{2}=M(\frac{1}{2}\ln{M}-1)\quad{\rm if}\ M=h^{2},\ {\rm in\ the\ case\
}\lambda=1.$

We can get this from Eqn. (19) for the case $M=h^{2}$ and $h^{2}>>h$. In this
sense, Zhang’s formula for $e$ is a validation of Eqn. (9) for $f(x)=M/x$ and
$M>>h$.

###  3.7 $h^{\prime}$-index

The $h$-index does not differentiate between authors whose ${\Theta_{\rm
head}}$ and ${\Theta_{\rm tail}}$ are very different. To reflect such
differences, Zhang defined another index $h^{\prime}$ [19], where

$h^{\prime}=\sqrt{\frac{{\Theta_{\rm head}}-h^{2}}{{\Theta_{\rm tail}}}}\ h.$

Using our approximations for ${\Theta_{\rm head}}$ and ${\Theta_{\rm tail}}$,
we get

$\displaystyle h^{\prime}$ $\displaystyle\approx\sqrt{\frac{{F_{\rm
head}}-h^{2}}{{F_{\rm tail}}}}\ h$
$\displaystyle=\sqrt{\frac{\frac{M}{M-h}\left(\ln{\frac{M}{h}}\right)-1}{\frac{N}{N-h}\left(\ln{\frac{N}{h}}\right)-1}}\
h$ (20)

Fig. 11 shows a regression line of gradient 0.94 for approximate $h^{\prime}$
values calculated with Eqn. (20) plotted against the empirical values. This is
a good approximation for expected $h^{\prime}$ value, but the data points are
quite dispersed, giving a correlation coefficient of $R^{2}\approx 0.86$.

Figure 11:  Comparing approximate value of $h^{\prime}$ in Eqn. (20) to
empirical value for the $h^{\prime}$-index.

For authors with $M>>h$ and $N>>h$, this simplifies to

$\displaystyle h^{\prime}$
$\displaystyle\approx\sqrt{\frac{\ln{\frac{M}{h}}-1}{\ln{\frac{N}{h}}-1}}\ h$
$\displaystyle=\sqrt{\frac{\ln{M}-\ln{{\tt e}h}}{\ln{N}-\ln{{\tt e}h}}}\ h$
(21)

(Note: It is not uncommon for authors to have $M<{\tt e}h$ or $N<{\tt e}h$,
for whom the assumption $M>>h$ or $N>>h$ is violated and the approximation
fails.)

To better understand the expression in Eqn. (21), consider a simplified
geometry using two triangles, as shown in Fig. 12, with areas

$\Delta_{\rm head}=\frac{1}{2}(M-h)h\quad{\rm and}\quad\Delta_{\rm
tail}=\frac{1}{2}(N-h)h.$

For this geometry, we get

$h^{\prime}=\sqrt{\frac{\Delta_{\rm head}}{\Delta_{\rm tail}}}\
h=\sqrt{\frac{M-h}{N-h}}\ h.$

Eqn. (21) shows how $h^{\prime}$ for this simplified geometry is modified when
we take into account the citation curvatures.

Figure 12:  Understanding Eqn. (21) with a simplified geometry.

###  3.8 $h_{2}$-index

In analyzing the citations for an author’s publications, one must first filter
out those by another author with a similar name. (E.g. The Computer Science
bibliography website DBLP lists more than 300 authors named “Wei Wang”.)

To reduce the effort needed to disambiguate authorship, Kosmulski defined a
$h_{2}$-index as the greatest integer such that the $h_{2}$ most-cited papers
have at least $h_{2}^{2}$ citations each. He observed that

$h_{2}\ {\displaystyle\propto}\ {\Theta}^{\frac{1}{3}}$ (22)

The focus of $h_{2}$ is in the head, so we can approximate $h_{2}$ by

$\displaystyle h_{2}^{2}$ $\displaystyle\approx{f_{\rm head}}(h_{2})$
$\displaystyle=\frac{{b_{\rm head}}}{h_{2}+{c_{\rm head}}}$
$\displaystyle=\frac{Mh^{2}}{h_{2}(M-h)+h^{2}}\quad{\rm by\
Eqn.~{}(\ref{eq:abchead})}$
$\displaystyle\approx\frac{Mh^{2}}{h_{2}M+h^{2}}\quad{\rm for\ }M>>h$

Therefore,

$\displaystyle h_{2}^{3}M$ $\displaystyle=h^{2}(M-h_{2}^{2})$
$\displaystyle{\rm i.e.}\quad h_{2}^{3}$
$\displaystyle=\frac{M-h_{2}^{2}}{M}h^{2}$ $\displaystyle{\rm so}\quad
h_{2}^{3}$ $\displaystyle\approx h^{2}\quad{\rm for\ }M>>h_{2}^{2}.$ (23)

This last approximation makes $h^{2}$ an over-estimate of $h_{2}^{3}$, as
indicated by the regression line (gradient 1.27) in Fig. 13.

Figure 13:  $h^{2}$ over-estimates $h_{2}^{3}$.

Eqn. (23) thus confirms Kosmulski’s observation (from a small dataset) that
$h_{2}^{3}=O(h^{2})$. Using Eqn. (8), we get

$h_{2}^{3}\approx\frac{F}{\ln\big{(}\frac{MN}{{\tt e}h^{2}}\big{)}}\quad{\rm
so\ }h_{2}\approx\Big{(}\frac{1}{\ln\big{(}\frac{MN}{{\tt
e}h^{2}}\big{)}}\Big{)}^{\frac{1}{3}}\Theta^{\frac{1}{3}}$

since $F$ is an approximation for $\Theta$. We thus see how Kosmulski’s
approximation (22) depends on $M$, $N$ and $h$.

###  3.9 $dc_{i}$ and $dc_{o}$: impact and potential

To measure the impact of an author’s publications, Silva and Grácio defined an
index [14]

$dc_{i}=\frac{1}{h}\sum_{n=1}^{h}(\Psi(n)-h),$

i.e. $dc_{i}=e^{2}/h$, using the $e$-index. From the approximation for $e$ in
Sec. 3.6, we get

$dc_{i}\approx h(\frac{M}{M-h}\ln\frac{M}{h}-1)$ (24)

Fig. 14(a) shows that the agreement between this approximation and the
empirical $dc_{i}$ values is weakened by dispersion ($R^{2}\approx 0.58$) and
outliers (gradient$\approx 0.76$). As with the $A$-index in Fig. 6, it is hard
to give an accurate formula for $dc_{i}$ since authors have large differences
in citation patterns for their highly-cited papers.

The motivation for $dc_{i}$ lies in differentiating two authors with similar
$h$ value. From Eqn. (24), we get

$dc_{i}\approx h\ln\frac{M}{{\tt e}h}\quad{\rm for\ }M>>h,$ (25)

and we see that the author with a larger $M$ indeed has a higher $dc_{i}$.
Silva and Grácio’s citation data for 116 Brazilian mathematicians has a
moderate Pearson correlation coefficient of 0.48 between $dc_{i}$ and $h$, and
we see this correlation in Eqn. (25) if $\ln\frac{M}{{\tt e}h}$ is considered
a multiplicative noise term for fixed $h$.

To measure the potential for increasing an author’s $h$ value, Silva and
Grácio defined an index

$dc_{o}=\frac{1}{N-h}\sum_{n>h}(h-\Psi(n)),$

i.e. $dc_{o}=h-{\Theta_{\rm tail}}/(N-h)$. Using ${F_{\rm tail}}$ in Eqn. (10)
to approximate ${\Theta_{\rm tail}}$, we get

$dc_{o}\approx
h-\frac{1}{N-h}\big{(}\frac{Nh^{2}}{N-h}\ln{(\frac{N}{h})}-h^{2}\big{)}$ (26)

Fig. 14(b) shows that — in contrast to Fig. 14(a) — there is excellent
agreement between this approximation and the empirical value (gradient$\approx
1.03$, $R^{2}\approx 0.97$).

Silva and Grácio argue that, for two authors with the same $h$ value, the one
with a larger $dc_{o}$ value has greater potential for increasing $h$. We can
see this by further approximating Eqn. (26):

$\displaystyle dc_{o}\approx h\big{(}1-\frac{h}{N}\ln(\frac{N}{{\tt
e}h})\big{)},$ (27)

where, for the same $h$, $dc_{o}$ is higher for the author with a larger $N$.
We can rewrite Eqn. (27) as

$dc_{o}\approx h-\frac{h^{2}}{N}\ln(\frac{N}{{\tt e}h}),$

so $dc_{o}$ is linear in $h$, with an additive noise term
$\frac{h^{2}}{N}\ln(\frac{N}{{\tt e}h})$ induced by $N$ and biased by $h$.
Indeed, Silva and Grácio’s data shows a strong linear correlation (Pearson
coefficient $\approx 0.96$) between $dc_{o}$ and $h$.

(a) Approximate vales using Eqn. (24). (b) Approximate vales using Eqn. (26).

Figure 14:  Comparing approximate and empirical values for $dc_{i}$ and
$dc_{o}$.

##  4 Considering time and group
activities

We now apply our approximations to analyze the effect of time and the
aggregation of citation counts.

###  4.1 Modeling the effect of time

In using $dc_{o}$ to measure the potential for increasing an author’s $h$
value, one can make a prediction. Supporting such a prediction requires some
model of how citations and publications increase over time.

When introducing the $h$-index, Hirsch gave a back-of-an-envelope derivation
that shows

$h=h_{0}t,$ (28)

where $t$ is the time since the author’s first publication, and $h_{0}$ is a
constant determined by publication rate and citation rate. There is some
empirical validation of Eqn. (28) [1, 5, 10, 12].

Burrell also provided numerical support using a stochastic model [6]. This
model assumes that the number of publications for an author is Poisson
distributed over time at a constant rate. One therefore expects

$N=N_{0}t,$ (29)

for some constant $N_{0}$. By a similar Poisson assumption, the number of
citations for a particular publication is expected to be linear with respect
to time. An author’s publications appear at different times and have different
citation rates (that are gamma distributed in Burrell’s model). The
publication with the highest citation — and the corresponding citation rate —
may therefore change over time. Even so, we further assume

$M=M_{0}t$ (30)

for some constant $M_{0}$. In the following, we refer to Eqns.(28)–(30) as the
linear model.

It follows from this model and Eqn. (4) that

$\displaystyle a$
$\displaystyle=\frac{M_{0}h_{0}^{2}}{M_{0}N_{0}-(M_{0}+N_{0})h_{0}}t$
$\displaystyle b$
$\displaystyle=M_{0}N_{0}(M_{0}-h_{0})(N_{0}-h_{0})\Big{(}\frac{h_{0}}{M_{0}N_{0}-(M_{0}+N_{0})h_{0}}\Big{)}^{2}t^{2}$
$\displaystyle c$
$\displaystyle=\frac{N_{0}h_{0}^{2}}{M_{0}N_{0}-(M_{0}+N_{0})h_{0}}t.$

Similarly, ${c_{\rm head}}$ and ${a_{\rm tail}}$ are linear in $t$, but
${b_{\rm head}}$ and ${b_{\rm tail}}$ are quadratic in $t$.

It follows from Eqn. (17) that

$g\approx\left(h_{0}\sqrt{\ln\Big{(}\frac{4M_{0}}{{\tt
e}h_{0}}\Big{)}}\right)t.$

Burrell observed this linearity in two numerical examples for his stochastic
model [7]. He pointed out that the correlation coefficient for $h_{2}$ vs $t$
is much smaller. In fact, we see from Eqn. (23) that

$h_{2}\approx h_{0}^{\frac{2}{3}}t^{\frac{2}{3}},$

so $h_{2}$ is not linear in $t$.

Burrell’s numerical examples also showed that ${\Theta_{\rm head}}$ and the
$A$-index are approximately proportional to $t^{2}$ and $t$ respectively.
Indeed, we see from Eqn. (9) and Eqn. (13) that

${\Theta_{\rm head}}\approx{F_{\rm
head}}\approx\big{(}\frac{M_{0}h_{0}^{2}}{M_{0}-h_{0}}\ln{\big{(}\frac{M_{0}}{h_{0}}\big{)}\big{)}}t^{2}$
(31)

and

$A\approx\left(\frac{M_{0}h_{0}}{M_{0}-h_{0}}\ln{\big{(}\frac{M_{0}}{h_{0}}\big{)}}\right)t,$

so the multiplicative factors are constant if $M,N$ and $h$ are linear in $t$.
Similarly, the factors

$\displaystyle\ln{\big{(}\frac{MN}{{\tt e}h^{2}}\big{)}}\quad$
$\displaystyle{\rm for\ }{\Theta}\ {\rm in\ Eqn.~{}(\ref{eq:ah2})}$
$\displaystyle\sqrt{\frac{M}{M-h}\ln{\Big{(}\frac{M}{h}\Big{)}}}\quad$
$\displaystyle{\rm for\ }R\ {\rm in\ Eqn.~{}(\ref{eq:Rindex})}$
$\displaystyle\sqrt[4]{\ln\big{(}\frac{4M}{{\tt e}h}\big{)}}\quad$
$\displaystyle{\rm for\ }\sqrt{hg}\ {\rm in\ Eqn.~{}(\ref{eq:hg})}$
$\displaystyle\frac{M}{M-h}\ln{\big{(}\frac{M}{h}\big{)}}-1\quad$
$\displaystyle{\rm for\ }e^{2}\ {\rm in\ Eqn.~{}(\ref{eq:eindex})}$
$\displaystyle\sqrt{\frac{\frac{M}{M-h}\left(\ln{\frac{M}{h}}\right)-1}{\frac{N}{N-h}\left(\ln{\frac{N}{h}}\right)-1}}\
h\quad$ $\displaystyle{\rm for\ }h^{\prime}\ {\rm in\
Eqn.~{}(\ref{eq:hprime})}$ $\displaystyle\frac{M}{M-h}\ln\frac{M}{h}-1\quad$
$\displaystyle{\rm for\ }dc_{i}\ {\rm in\ Eqn.~{}(\ref{eq:dci1})}$
$\displaystyle
1-\frac{h}{N-h}\big{(}\frac{N}{N-h}\ln{(\frac{N}{h})}-1\big{)}\quad$
$\displaystyle{\rm for\ }dc_{o}\ {\rm in\ Eqn.~{}(\ref{eq:dco1})}$

are constants in the linear model.

One issue with the $h$-index is that it provides no information for
distinguishing two authors with the same integer value $h$. Even when they are
different, authors may have larger $h$ values because they have been
publishing for a longer time. Hirsch himself recommended using $h/t$, i.e.
$h_{0}$ in the linear model, to compare authors with different seniority.

As mentioned above, Burrell used his probabilistic model to examine how
${\Theta_{\rm head}}$ varies with time. If the linear model holds for our
approximation, then

$\displaystyle\frac{d{\Theta_{\rm head}}}{dt}$ $\displaystyle\approx
2\Big{(}\frac{M_{0}h_{0}^{2}}{M_{0}-h_{0}}\ln{\Big{(}\frac{M_{0}}{h_{0}}\Big{)}}\Big{)}t\quad{\rm
from\ Eqn.~{}(\ref{eq:t2})}$
$\displaystyle=\frac{2}{t}\Big{(}\frac{Mh^{2}}{M-h}\ln{\Big{(}\frac{M}{h}\Big{)}}\Big{)}$
$\displaystyle\approx 2\frac{{\Theta_{\rm head}}}{t}$

We see that (like using $h/t$ to differentiate two authors with the same $h$)
for two authors with the same ${\Theta_{\rm head}}$, the senior author has a
larger $t$ and thus a smaller growth rate $\frac{d{\Theta_{\rm head}}}{dt}$.

###  4.2 $h$-index for a group

The concept of $h$-index has been extended from an individual author to a
group (department [15], journal [13], etc.). Here, we apply our equation to
derive the $h$-index of a group from the individual $h$-indices.

Consider a group of $r$ authors. Let $M_{i}$, $N_{i}$ and $h_{i}$ be the $M$,
$N$ and $h$ values for the $i$-th author in the group, and $M_{*}$, $N_{*}$
and $h_{*}$ the $M$, $N$ and $h$ values for the collection of publications
from this group. Then,

$M_{*}=\max\\{M_{1},\ldots,M_{r}\\}.$ (32)

For a first approximation, we assume no two authors in the group share a
publication, so

$N_{*}=N_{1}+\cdots+N_{r}.$ (33)

Suppose the $i$-th author has $x_{i}$ publications with at least $h_{*}$
citations each. By the definition of the $h$-index,

$h_{*}=x_{1}+\cdots+x_{r}.$ (34)

Since $x_{i}\leq h_{i}$ (see Fig. 15), we can use ${f_{\rm head}}$ to
approximate the citation data for each author. Let $a_{i}$, $b_{i}$ and
$c_{i}$ be the ${a_{\rm head}}$, ${b_{\rm head}}$ and ${c_{\rm head}}$ values
for the $i$-th author. By Eqn. (5),

$\displaystyle\begin{split}a_{i}&=0\\\
b_{i}&=\frac{M_{i}h_{i}^{2}}{M_{i}-h_{i}}\\\
c_{i}&=\frac{h_{i}^{2}}{M_{i}-h_{i}}\\\ {\rm and}\quad
f_{i}(x)&=\frac{b_{i}}{x+c_{i}}\ ,\end{split}$ (35)

where $f_{i}$ is ${f_{\rm head}}$ for the $i$-th author. Then

$h_{*}=\frac{b_{i}}{x_{i}+c_{i}}\quad{\rm for\ }i=1,\ldots,r,$

so

$\sum_{i=1}^{r}h_{*}(x_{i}+c_{i})=\sum_{i=1}^{r}b_{i}.$

By Eqn. (34),

$h_{*}^{2}+h_{*}\sum_{i=1}^{r}c_{i}-\sum_{i=1}^{r}b_{i}=0.$

Thus

$h_{*}=\frac{-\sum_{i=1}^{r}c_{i}+\sqrt{\big{(}\sum_{i=1}^{r}c_{i}\big{)}^{2}+4(\sum_{i=1}^{r}b_{i})}}{2}$
(36)

Note that $b_{i}=M_{i}c_{i}$ and $M_{i}>>1$ for most authors. For
$4\sum_{i=1}^{r}b_{i}>>(\sum_{i=1}^{r}c_{i})^{2}$, we can use the following
approximation:

$\displaystyle h_{*}$
$\displaystyle\approx\frac{-\sum_{i=1}^{r}c_{i}+\sqrt{4(\sum_{i=1}^{r}b_{i})}}{2}$
$\displaystyle\approx\sqrt{\sum_{i=1}^{r}b_{i}}$
$\displaystyle=\sqrt{\sum_{i=1}^{r}\frac{M_{i}h_{i}^{2}}{M_{i}-h_{i}}}.$ (37)

To validate this approximation, we selected 9 authors from our dataset. Table
1 lists, for author $i$, the research area, $M_{i}$, $N_{i}$ and $h_{i}$. For
$r=2,\ldots,9$, we grouped the first $r$ authors’ publications to determine
the empirical $h_{*}$ value for the group.

Figure 15: $h_{i}\leq h_{*}$, so $x_{i}\leq h_{*}$.

$i$ | research area | $M_{i}$ | $N_{i}$ | $h_{i}$
---|---|---|---|---
1 | physics | 336 | 15 | 13
2 | city planning | 423 | 90 | 27
3 | public health | 2108 | 63 | 32
4 | physiology | 1161 | 34 | 18
5 | computer science | 262 | 396 | 44
6 | public policy | 364 | 128 | 31
7 | sociology | 901 | 64 | 24
8 | psychology | 272 | 124 | 46
9 | artificial intelligence | 513 | 94 | 19

| $h_{*}$
---|---
$r$ | empirical value | approximation (Eqn. (37))
2 |  32 | 30.9
3 |  49 | 44.7
4 |  57 | 48.2
5 |  71 | 68.2
6 |  77 | 75.5
7 |  92 | 79.3
8 | 103 | 94.0
9 | 105 | 96.0

Table 1:  Comparing approximate value of $h_{*}$ in Eqn. (37) to the empirical
$h_{*}$.

Note that the approximation (37) takes into account the $M_{i}$ value for each
author in the collection when estimating the aggregate $h_{*}$ value. The
omission of $\sum_{i=1}^{r}c_{i}$ leads to an under-estimation, but Fig. 16
shows that it nonetheless gives a good approximation for $h_{*}$ (the
regression line has gradient 0.91 and $R^{2}\approx 0.98$).

Figure 16:  Comparing approximate value of $h_{*}$ in Eqn. (37) to the
empirical $h_{*}$.

##  5 Conclusion

In this paper, we proposed a simple equation to approximate the citation count
distribution of an author. The equation is based on the idea of using 3
geometrical characteristics ($M$, $N$ and $h$) of the count distribution to
calibrate 3 parameters ($a$, $b$ and $c$) for an equation to approximate the
distribution.

We demonstrated the equation’s usefulness in the analysis of such
distributions by deriving closed-form expressions for various citation
indices, and using them to model the effect of time, and identify individual
contribution to a group $h$-index.

## References

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## Appendix

The following histograms describe the sample of 226 authors:


# Convergence analysis on the alternating direction method of multipliers for
the cosparse optimization problem

Zisheng Liu a and Ting Zhang b CONTACT Zisheng Liu Email: Z. Liu Email:
<EMAIL_ADDRESS>aSchool of Statistics and Big Data, Henan University of
Economics and Law, Zhengzhou, China;
bSchool of Mathematics and Information Science, Henan University of Economics
and Law, Zhengzhou, China.

###### Abstract

From a dual perspective of the sparse representation model, Nam et al.
proposed the cosparse analysis model. In this paper, we aim to investigate the
convergence of the alternating direction method of multipliers (ADMM) for the
cosparse optimization problem. First, we examine the variational inequality
representation of the cosparse optimization problem by introducing auxiliary
variables. Second, ADMM is used to solve cosparse optimization problem.
Finally, by utilizing a tight frame with a uniform row norm and building upon
lemmas and the strict contraction theorem, we establish a worst-case
$\mathcal{O}(1/t)$ convergence rate in the ergodic sense.

###### keywords:

sparse representation model, cosparse analysis model, alternating direction
method of multipliers, variational inequality, convergence analysis

90C25

##  1 Introduction

Low-dimensional signal recovery takes advantage of the inherent low-
dimensionality of many natural signals, despite their high ambient dimension.
Utilizing prior information about the low-dimensional space can significantly
aid in recovering the signal of interest. Sparsity, a widely recognized form
of prior information, serves as the foundation for the burgeoning field of
compressive sensing (CS [1, 2, 3, 4]). The recovery of sparse inputs has found
numerous applications in areas such as imaging, speech, radar signal
processing, sub-Nyquist sampling, and more [5, 6, 7, 8]. A typical sparse
recovery problem is associated with the following linear system:

$\displaystyle y=Mx,$ (1)

where $y\in R^{m}$ is an observed vector, $M\in R^{m\times d}$ is a
measurement matrix and $x\in R^{d}$ is an unknown signal which would be
estimated from $y$. According to the Nyquist-Shannon sampling theorem, if the
$k$-space data is undersampled so much that it fails to meet the Nyquist
sampling criterion, then reconstructing the data can be difficult or
impossible without prior knowledge of $x$.

###  1.1 Sparse synthesis model

Over the past decade, the application of compressed sensing significantly
increased the image reconstruction speed and efficiency because of its
capability to reconstruct images from highly undersampled signals. Sparse
prior is widely used in CS-based reconstruction methods. For the sparse
synthesis model, if a vector $x$ is sufficient sparse, under the incoherence
assumptions on the measurement matrix $M$, $x$ can be robustly estimated by
the problem

$\displaystyle\min_{x}$ $\displaystyle~{}\|x\|_{\tau}$ (2) $\displaystyle
s.t.$ $\displaystyle~{}y=Mx,$

where $0\leq\tau\leq 1$. The advanced ideas and methods have been explored by
applications in signals and image processing [9, 10, 11, 12]. After years of
research, this model is becoming more and more mature and stable.

###  1.2 Cosparse analysis model

In the recent decade, the cosparse analysis model is an alternative approach
has gained popularity [13, 14, 15, 16]. Within this framework, a potentially
redundant analysis operator $D\in\mathbb{R}^{n\times d}(n\geq d)$ is employed,
and the analyzed vector $Dx$ is expected to be sparse. This implies that a
signal $x\in R^{d}$ belongs to the cosparse analysis model with cosparsity
$\ell$ if $\ell=n-\|Dx\|_{0}$. In this paper, the quantity $\ell$ represents
the number of rows in $D$ that are orthogonal to the signal. Consequently, $x$
is referred to as $\ell$-cosparse or simply cosparse. The specific definitions
of cosparse and cosupport can be found in literature [14], for ease of
reference, we have listed them below.

######  Definition 1.1 (Cosparse).

A signal $x\in R^{d}$ is said to be cosparse with respect to an analysis
operator $D\in R^{n\times d}$ if the analysis representation vector $Dx$
contains many zero elements. Further, the number of zero elements

$\displaystyle\ell=n-\|Dx\|_{0}$

is called the cosparsity of $x$, we also say $x$ is $\ell$-cosparse.

######  Definition 1.2 (Cosupport).

For a signal $x\in R^{d}$ and a given analysis operator $D\in R^{n\times d}$
with its rows $D_{j}\in R^{d}(1\leq j\leq n)$, the cosupport is defined by

$\displaystyle\Lambda:=\\{j|\langle D_{j},x\rangle=0\\}.$

In this paper, $D$ is a tight frame with uniform row norm. We remind the
reader that a frame is defined as below.

######  Definition 1.3 (Frame[17, 18]).

Let $\Phi=\\{\varphi_{i}\\}_{i=1}^{N}\subseteq R^{n}$ be a vector sequence of
the Hilbert space with $N\geq n$. If there exist constants $0<A\leq B<\infty$
such that

$\displaystyle\forall x\in R^{n},~{}~{}~{}A\|x\|^{2}\leq\sum_{i=1}^{N}|\langle
x,\varphi_{i}\rangle|^{2}\leq B\|x\|^{2},$ (3)

then $\Phi$ is referred to as a finite frame of $R^{n}$. The constants $A$ and
$B$ in the above formula are known as the lower and upper bounds of the finite
frame $\Phi$, respectively. They are considered to be the optimal bounds, with
$A$ being the supremum in the lower bound and $B$ being the infimum in the
upper bound. If $A=B$, then the frame $\Phi$ is called an $A$-tight frame. If
$A=B=1$, then $\Phi$ is called a Parseval frame. If there exists a constant
$C$ such that each meta-norm $\|\varphi_{i}\|=C$ of the frame $\Phi$, then
$\Phi$ is called an iso-norm frame. In particular, for a tight frame, if
$C=1$, it is referred to as a uniformly tight frame.

According to the definition of cosparsity, the cosparse analysis model focuses
on the zero elements of the analysis representation vector $Dx$, rather than
the non-zero elements. This perspective contrasts with the sparse synthesis
model. If the cosparsity $\ell$ is significantly large, meaning that the
number of zeros $\ell$ is close to $d$, we say that $x$ has a cosparse
representation. The cosupport set is identified by iteratively removing rows
from $D$ for which $\langle D_{j},x\rangle\neq 0$ until the index set
$\Lambda$ remains unchanged, with $|\Lambda|\geq\ell$.

If the analysis representation vector $Dx$ is sparse, similar to the sparse
model, the estimation of $x$ from the measurements can be achieved by

$\displaystyle\min_{x}$ $\displaystyle~{}\|Dx\|_{0}$ (4) $\displaystyle s.t.$
$\displaystyle~{}y=Mx.$

The minimization problem (4) is known to be NP-hard [14], necessitating the
use of approximation methods. Similar to the sparse model, one option is to
use the greedy analysis pursuit (GAP) approach, which is inspired by the
orthogonal matching pursuit (OMP) algorithm [13, 14]. Alternatively, the
nonconvex $\ell_{0}$ norm can be approximated by the convex $\ell_{1}$ norm,
leading to the relaxed problem known as analysis basis pursuit (ABP) [19]. In
this case, $x$ can be estimated by solving a modified optimization problem

$\displaystyle\min_{x}\|Dx\|_{1}$ (5) $\displaystyle
s.t.~{}\|y-Mx\|_{2}\leq\epsilon,$

where $\|\cdot\|_{1}$ is the $\ell_{1}$ norm that sums the absolute values of
a vector and $\epsilon$ is a upper bound on the noise level $\|v\|_{2}$.

ABP is equivalent to the unconstrained optimization

$\displaystyle\min_{x}\|Dx\|_{1}+\frac{\alpha}{2}\|y-Mx\|^{2}_{2},$ (6)

which we call analysis LASSO (ALASSO). It can be said that ABP and ALASSO are
equivalent in the sense that for any $\epsilon>0$, there exists an $\alpha$
such that the optimal solutions of ABP and ALASSO are identical. For the
optimization problem (6), our previous work presented the modified GAP
algorithm and error analysis [20, 16]. The simulations we conducted
demonstrated the advantages of the proposed method for the cosparse
optimization problem. These optimization problems can also be solved using
interior point methods [21]. However, as the problem dimension increases,
these techniques become time-consuming since they require solutions of linear
systems. Other suggested approaches include the alternating direction method
of multipliers (ADMM) [22, 23, 24, 33] and the accelerated alternating
minimization method (AAM) [25]. In this paper, we propose a new way to analyze
the convergence theory of the cosparse optimization problem based on the
variational inequality.

###  1.3 Organization of the paper

Our focus in this paper is on the cosparse optimization problem and its
convergence study based on a variational inequality. The paper is structured
as follows: In Section 2, we introduce auxiliary variables and investigate the
variational inequality characterization of the cosparse optimization problem.
In Section 3, we present several lemmas that establish the strict contraction
of the ADMM for the cosparse optimization problem. Using these lemmas and the
strict contraction theorem, we establish a worst-case $\mathcal{O}(1/t)$
convergence rate in the ergodic sense. Finally, Section 4 provides a brief
conclusion.

##  2 Preliminaries

To apply the ADMM for solving the cosparse optimization problem (6), we
convert the unconstrained optimization problem mentioned above into a
constrained optimization problem as follows

$\displaystyle\min_{x,z}$
$\displaystyle\|z\|_{1}+\frac{\alpha}{2}\|y-Mx\|^{2}_{2}$ (7) $\displaystyle
s.t.$ $\displaystyle~{}Dx-z=0,$

where an auxiliary variable $z\in R^{n}$ is introduced in (6) to transfer $Dx$
out of the nondifferentiable term $\|\cdot\|_{1}$ and $\alpha>0$ is a penalty
parameter.

In this section, we summarize the variational inequality (VI) characterization
of (7). Initially, we present the optimality condition of the constrained
optimization problem (7), which forms the foundation for our subsequent
convergence analysis [26, 28]. We then proceed to express the Lagrangian
function of (7) as follows

$\displaystyle
L(z,x,\lambda)=\|z\|_{1}+\frac{\alpha}{2}\|y-Mx\|^{2}_{2}-\lambda^{T}(Dx-z).$
(8)

In (8), we assume that $x\in\mathcal{X}$, $z\in\mathcal{Z}$ and $\lambda\in
R^{n}$ where $\mathcal{X}\subset R^{d}$ and $\mathcal{Z}\subset R^{n}$ are
closed convex sets, we call
($z^{*},x^{*},\lambda^{*})\in\Omega:=\mathcal{Z\times X}\times R^{n}$ to be a
saddle point of $L(z,x,\lambda)$ if the following inequalities are satisfied

$\displaystyle\begin{aligned} L(z^{*},x^{*},\lambda)&\leq
L(z^{*},x^{*},\lambda^{*})\leq L(z,x,\lambda^{*}).\end{aligned}$ (9)

Obviously, a saddle point ($z^{*},x^{*},\lambda^{*}$) can be characterized by
the system

$\displaystyle\left\\{\begin{array}[]{lll}z^{*}=\arg\min\\{L(z,x^{*},\lambda^{*})|z\in\mathcal{Z}\\},\\\
x^{*}=\arg\min\\{L(z^{*},x,\lambda^{*})|x\in\mathcal{X}\\},\\\
\lambda^{*}=\arg\max\\{L(z^{*},x^{*},\lambda)|\lambda\in
R^{n}\\},\end{array}\right.$ (13)

which can be rewritten as

$\displaystyle\begin{aligned}
\left\\{\begin{array}[]{lll}z^{*}\in\mathcal{Z},L(z,x^{*},\lambda^{*})-L(z^{*},x^{*},\lambda^{*})\geq
0,\\\
x^{*}\in\mathcal{X},L(z^{*},x,\lambda^{*})-L(z^{*},x^{*},\lambda^{*})\geq
0,\\\ \lambda^{*}\in
R^{n},L(z^{*},x^{*},\lambda^{*})-L(z^{*},x^{*},\lambda)\geq
0.\end{array}\right.\end{aligned}$ (14)

Below, we present a summary of the method for expressing the optimality
condition of the cosparse analysis model (7) via a variational inequality.

######  Proposition 2.1.

Suppose $\mathcal{X}\subset R^{d}$ is a closed convex set, and
$\theta(x):R^{d}\rightarrow R$ is a convex function. Furthermore, let $f(x)$
be differentiable in $\mathcal{X}$. We assume that the set of solutions for
the minimization problem $\min\\{\theta(x)+f(x)|x\in\mathcal{X}\\}$ is
nonempty, then,

$\displaystyle x^{*}=\arg\min\\{\theta(x)+f(x)|x\in\mathcal{X}\\}$ (15)

if and only if

$\displaystyle
x,~{}x^{*}\in\mathcal{X},~{}\theta(x)-\theta(x^{*})+(x-x^{*})^{T}\nabla
f(x^{*})\geq 0.$ (16)

The proof of Proposition 2.1 is available in [29]. Let
$\theta_{1}(z)=\|z\|_{1}$ and
$\theta_{2}(x)=\frac{\alpha}{2}\|y-Mx\|^{2}_{2}$, according to the above
inequality (16), a saddle point ($z^{*},x^{*},\lambda^{*}$) of the Lagrangian
function (8) can be characterized by a solution point of the following
variational inequality

$\displaystyle\begin{aligned}
&\omega,\omega^{*}\in\Omega,~{}~{}\theta(u)-\theta(u^{*})+(\omega-\omega^{*})^{T}F(\omega^{*})\geq
0,\end{aligned}$ (17)

where

$\displaystyle\theta(u)=\theta_{1}(z)+\theta_{2}(x),~{}~{}\Omega=\mathcal{Z\times
X}\times R^{n},$ (18)

and

$\displaystyle\begin{aligned} &\omega=\left(\begin{array}[]{c}z\\\ x\\\
\lambda\\\ \end{array}\right),~{}~{}u=\left(\begin{array}[]{c}z\\\ x\\\
\end{array}\right),~{}~{}F(\omega)=\left(\begin{array}[]{c}\lambda\\\
-D^{T}\lambda\\\ Dx-z\\\ \end{array}\right),\end{aligned}$ (19)

Since $F$ is an affine operator, and

$\displaystyle F(\omega)=\left(\begin{array}[]{ccc}0&0&I\\\ 0&0&-D^{T}\\\
-I&D&0\\\ \end{array}\right)\left(\begin{array}[]{c}z\\\ x\\\ \lambda\\\
\end{array}\right),$ (26)

According to the antisymmetry of the affine matrix, it follows that

$\displaystyle(\omega-\bar{\omega})^{T}[F(\omega)-F(\bar{\omega})]\equiv
0,~{}\forall~{}\omega,\bar{\omega}\in\Omega.$ (27)

Using inequality (16) and combining (8), we derive the following conclusion
with $(z^{*},x^{*},\lambda^{*})\in\Omega$,

$\displaystyle\left\\{\begin{array}[]{lll}\theta_{1}(z)-\theta_{1}(z^{*})+(z-z^{*})^{T}\lambda^{*}\geq
0,\\\ \theta_{2}(x)-\theta_{2}(x^{*})+(x-x^{*})^{T}(-D^{T}\lambda^{*})\geq
0,\\\ (\lambda-\lambda^{*})^{T}(Dx^{*}-z^{*})\geq 0.\end{array}\right.$ (31)

After conducting the aforementioned analysis, the linear constrained cosparse
optimization problem is reformulated as a variational inequality.
Consequently, the task is ultimately simplified to identifying a saddle point
of the Lagrangian function. In the subsequent section, the convergence
analysis of the ADMM method for addressing the cosparse optimization problem,
as denoted by equation (7), will be discussed.

##  3 Convergence analysis of the cosparse optimization problem

###  3.1 Variational inequality characterization of ADMM

The augmented Lagrangian function of the problem (7) can be formulated as
follows

$\displaystyle\begin{aligned}
\mathcal{L}_{\beta}(z,x,\lambda)=&\|z\|_{1}+\frac{\alpha}{2}\|y-Mx\|^{2}_{2}-\lambda^{T}(Dx-z)+\frac{\beta}{2}\|Dx-z\|_{2}^{2},\end{aligned}$
(32)

where $\lambda$ is the Lagrange multiplier and $\beta>0$ is a penalty
parameter for the linear constraints. Thus, applying directly the augmented
Lagrangian function (32) and starting with an initial iterate
$(x^{0},\lambda^{0})\in\mathcal{X}\times R^{n}$, the ADMM generates its
sequence via following iterative scheme

$\displaystyle\left\\{\begin{array}[]{lll}z^{k+1}=\arg\min\\{\mathcal{L}_{\beta}(z,x^{k},\lambda^{k})|z\in\mathcal{Z}\\},\\\
x^{k+1}=\arg\min\\{\mathcal{L}_{\beta}(z^{k+1},x,\lambda^{k})|x\in\mathcal{X}\\},\\\
\lambda^{k+1}=\lambda^{k}-\beta(Dx^{k+1}-z^{k+1}),~{}\lambda\in
R^{n}\end{array}\right.$ (36)

the corresponding variational inequalities of (36) can be given as

$\displaystyle\begin{aligned}
\left\\{\begin{array}[]{lll}\theta_{1}(z)-\theta_{1}(z^{k+1})+(z-z^{k+1})^{T}[\lambda^{k}-\beta(Dx^{k}-z^{k+1})]\geq
0,\\\
\theta_{2}(x)-\theta_{2}(x^{k+1})+(x-x^{k+1})^{T}[-D^{T}\lambda^{k}+\beta
D^{T}(Dx^{k+1}-z^{k+1})]\geq 0,\\\
(\lambda-\lambda^{k+1})^{T}[(Dx^{k+1}-z^{k+1})+\frac{1}{\beta}(\lambda^{k+1}-\lambda^{k})]\geq
0.\end{array}\right.\end{aligned}$ (37)

For some reviews on the classical ADMM, one can refer to literatures [24, 30,
31, 32, 26, 27].

###  3.2 Assertions

To establish that $\\{\omega^{k}\\}$ is strictly contractive with respect
$\Omega$, we first present several lemmas.

######  Lemma 3.1.

Let the sequence $\\{\omega^{k}\\}$ be generated by (36). Then, we have

$\displaystyle\theta(u)-\theta(u^{k+1})+(\omega-\omega^{k+1})^{T}F(\omega)$
(38) $\displaystyle\geq$
$\displaystyle(z-z^{k+1})^{T}\beta(Dx^{k}-Dx^{k+1})+\frac{1}{\beta}(\lambda-\lambda^{k+1})^{T}(\lambda^{k}-\lambda^{k+1}),~{}\forall\omega\in\Omega.$

###### Proof.

From (37) we know that

$\displaystyle\theta_{1}(z)-\theta_{1}(z^{k+1})+(z-z^{k+1})^{T}[\lambda^{k}-\beta(Dx^{k}-z^{k+1})]\geq
0,~{}\forall z\in\mathcal{Z}$ (39)

and

$\displaystyle\begin{aligned}
\theta_{2}(x)-\theta_{2}(x^{k+1})+(x-x^{k+1})^{T}(-D^{T}\lambda^{k}+\beta
D^{T}(Dx^{k+1}-z^{k+1}))\geq 0,~{}\forall x\in\mathcal{X}.\end{aligned}$ (40)

Using $\lambda^{k+1}=\lambda^{k}-\beta(Dx^{k+1}-z^{k+1})$ we can easily deduce

$\displaystyle\lambda^{k}=\lambda^{k+1}+\beta(Dx^{k+1}-z^{k+1})$ (41)

and

$\displaystyle(Dx^{k+1}-z^{k+1})=\frac{1}{\beta}(\lambda^{k}-\lambda^{k+1}).$
(42)

Putting the formulations (41) and (42) into (39) and (40), respectively, then
we have the following inequalities

$\displaystyle\theta_{1}(z)-\theta_{1}(z^{k+1})+(z-z^{k+1})^{T}[\lambda^{k+1}+\beta(Dx^{k+1}-z^{k+1})-\beta(Dx^{k}-z^{k+1})]\geq
0,$ (43)
$\displaystyle\theta_{2}(x)-\theta_{2}(x^{k+1})+(x-x^{k+1})^{T}(-D^{T}\lambda^{k+1})\geq
0,$ (44)

and

$\displaystyle\begin{aligned}
(\lambda-\lambda^{k+1})^{T}(Dx^{k+1}-z^{k+1})\geq(\lambda-\lambda^{k+1})^{T}\frac{1}{\beta}(\lambda^{k}-\lambda^{k+1}).\end{aligned}$
(45)

Combining (43), (44) and (45) we have

$\displaystyle\begin{aligned}
\left\\{\begin{array}[]{lll}\theta_{1}(z)-\theta_{1}(z^{k+1})+(z-z^{k+1})^{T}\lambda^{k+1}\geq(z-z^{k+1})^{T}\beta(Dx^{k}-Dx^{k+1}),\\\
\theta_{2}(x)-\theta_{2}(x^{k+1})+(x-x^{k+1})^{T}(-D^{T}\lambda^{k+1})\geq
0,\\\
(\lambda-\lambda^{k+1})^{T}(Dx^{k+1}-z^{k+1})\geq(\lambda-\lambda^{k+1})^{T}\frac{1}{\beta}(\lambda^{k}-\lambda^{k+1}),\end{array}\right.\end{aligned}$
(46)

which is

$\displaystyle\theta(u)-\theta(u^{k+1})+(\omega-\omega^{k+1})^{T}F(\omega^{k+1})$
$\displaystyle\geq$
$\displaystyle(z-z^{k+1})^{T}\beta(Dx^{k}-Dx^{k+1})+(\lambda-\lambda^{k+1})^{T}\frac{1}{\beta}(\lambda^{k}-\lambda^{k+1}).$

Note that the matrix in the operator $F$ is skew-symmetric, then, using (27),
we have

$\displaystyle\theta(u)-\theta(u^{k+1})+(\omega-\omega^{k+1})^{T}F(\omega)$
(47) $\displaystyle\geq$
$\displaystyle(z-z^{k+1})^{T}\beta(Dx^{k}-Dx^{k+1})+(\lambda-\lambda^{k+1})^{T}\frac{1}{\beta}(\lambda^{k}-\lambda^{k+1}).$

The Lemma 3.1 is proved. ∎

######  Lemma 3.2.

Let the sequence $\\{\omega^{k}\\}$ be generated by (36). Then, we have

$\displaystyle\beta(z-z^{k+1})^{T}(Dx^{k}-Dx^{k+1})+\frac{1}{\beta}(\lambda-\lambda^{k+1})^{T}(\lambda^{k}-\lambda^{k+1})$
(48) $\displaystyle=$
$\displaystyle-\frac{1}{2\beta}\|\lambda^{k}-\lambda\|_{2}^{2}-\frac{\beta}{2}\|Dx^{k}-z\|_{2}^{2}+\frac{1}{2\beta}\|\lambda^{k+1}-\lambda\|_{2}^{2}+\frac{\beta}{2}\|Dx^{k+1}-z\|_{2}^{2}$
$\displaystyle+\frac{\beta}{2}\|Dx^{k}-z^{k+1}\|_{2}^{2}.$

###### Proof.

Applying the identity

$\displaystyle(a-b)^{T}(c-d)=\frac{1}{2}\\{\|a-d\|_{2}^{2}-\|a-c\|_{2}^{2}\\}+\frac{1}{2}\\{\|c-b\|_{2}^{2}-\|d-b\|_{2}^{2}\\}$

to the left-hand side in (48) with

$a=z,~{}b=z^{k+1},~{}c=Dx^{k},~{}d=Dx^{k+1},$

we obtain

$\displaystyle\beta(z-z^{k+1})^{T}(Dx^{k}-Dx^{k+1})$ (49) $\displaystyle=$
$\displaystyle\frac{\beta}{2}\\{\|z-Dx^{k+1}\|_{2}^{2}-\|z-Dx^{k}\|_{2}^{2}\\}+\frac{\beta}{2}\\{\|Dx^{k}-z^{k+1}\|_{2}^{2}-\|Dx^{k+1}-z^{k+1}\|_{2}^{2}\\}.$

Using the identity

$b^{T}(b-a)=\frac{1}{2}(\|b\|_{2}^{2}-\|a\|_{2}^{2}+\|b-a\|_{2}^{2}),$

and let

$a=\lambda-\lambda^{k},~{}b=\lambda-\lambda^{k+1},$

we obtain

$\displaystyle\frac{1}{\beta}(\lambda-\lambda^{k+1})^{T}(\lambda^{k}-\lambda^{k+1})=\frac{1}{2\beta}\\{\|\lambda-\lambda^{k+1}\|_{2}^{2}-\|\lambda-\lambda^{k}\|_{2}^{2}+\|\lambda^{k}-\lambda^{k+1}\|_{2}^{2}\\}.$
(50)

Using

$\beta\|Dx^{k+1}-z^{k+1}\|_{2}^{2}=\frac{1}{\beta}\|\lambda^{k}-\lambda^{k+1}\|_{2}^{2},$

and combining (49) and (50), we complete the proof of this lemma. ∎

######  Lemma 3.3.

Let the sequence $\\{x^{k}\\}$, $\\{z^{k}\\}$ and $\\{\lambda^{k}\\}$ be
generated by (36), then,

$\displaystyle\begin{aligned}
\beta\|Dx^{k}-z^{k+1}\|_{2}^{2}\geq\beta\|Dx^{k}-Dx^{k+1}\|_{2}^{2}+\frac{1}{\beta}\|\lambda^{k}-\lambda^{k+1}\|_{2}^{2}.\end{aligned}$
(51)

###### Proof.

Based on the second inequality of inequality (46), we can derive the following
result

$\displaystyle\left\\{\begin{array}[]{ll}\theta_{2}(x)-\theta_{2}(x^{k+1})+(x-x^{k+1})^{T}(-D^{T}\lambda^{k+1})\geq
0,\\\ \theta_{2}(x)-\theta_{2}(x^{k})+(x-x^{k})^{T}(-D^{T}\lambda^{k})\geq
0.\end{array}\right.$ (54)

Let $x=x^{k}$ and $x=x^{k+1}$ in (54), respectively, then

$\displaystyle\left\\{\begin{array}[]{ll}\theta_{2}(x^{k})-\theta_{2}(x^{k+1})+(x^{k}-x^{k+1})^{T}(-D^{T}\lambda^{k+1})\geq
0,\\\
\theta_{2}(x^{k+1})-\theta_{2}(x^{k})+(x^{k+1}-x^{k})^{T}(-D^{T}\lambda^{k})\geq
0.\\\ \end{array}\right.$

From above inequalities, we have

$\displaystyle(\lambda^{k}-\lambda^{k+1})^{T}(Dx^{k}-Dx^{k+1})\geq 0.$ (56)

Using

$(Dx^{k+1}-z^{k+1})=\frac{1}{\beta}(\lambda^{k}-\lambda^{k+1}),$

then we obtain

$\displaystyle\beta\|Dx^{k}-z^{k+1}\|_{2}^{2}$ (57) $\displaystyle=$
$\displaystyle\beta\|Dx^{k}-Dx^{k+1}+Dx^{k+1}-z^{k+1}\|_{2}^{2}$
$\displaystyle=$
$\displaystyle\beta\|Dx^{k}-Dx^{k+1}+\frac{1}{\beta}(\lambda^{k}-\lambda^{k+1})\|_{2}^{2}$
$\displaystyle\geq$
$\displaystyle\beta\|Dx^{k}-Dx^{k+1}\|_{2}^{2}+\frac{1}{\beta}\|\lambda^{k}-\lambda^{k+1})\|_{2}^{2}.$

The proof of this lemma is completed. ∎

###  3.3 Strict contraction

To present the main result of the paper, it is necessary to establish the
strict contractility of the iterative sequence. The following subsection
provides a proof of the strong contractility of the iterative sequence
$\\{\omega^{k}\\}$, which relies on Lemma 3.1, Lemma 3.2, and Lemma 3.3.

######  Theorem 3.4.

Assuming that the sequence $\\{\omega^{k}\\}$ is generated by equation (36),
we can state the following

$\displaystyle\|v^{k+1}-v^{*}\|_{H}^{2}\leq\|v^{k}-v^{*}\|_{H}^{2}-\|v^{k}-v^{k+1}\|_{H}^{2}$
(58)

where

$\displaystyle\begin{aligned} v=\left(\begin{array}[]{c}\lambda\\\ x\\\
\end{array}\right),~{}~{}H=\left(\begin{array}[]{cc}\frac{1}{\beta}I_{m}&0\\\
0&\beta I_{d}\\\
\end{array}\right),~{}~{}\mathcal{V}^{*}=\\{(\lambda^{*},x^{*})|(z^{*},x^{*},\lambda^{*})\in\Omega\\}.\end{aligned}$
(59)

###### Proof.

We can deduce from Lemma 3.1 and Lemma 3.2 that

$\displaystyle\theta(u^{k+1})-\theta(u)+(\omega^{k+1}-\omega)^{T}F(\omega)$
(60) $\displaystyle\leq$
$\displaystyle\frac{1}{2\beta}\|\lambda^{k}-\lambda\|_{2}^{2}+\frac{\beta}{2}\|Dx^{k}-z\|_{2}^{2}-\frac{1}{2\beta}\|\lambda^{k+1}-\lambda\|_{2}^{2}-\frac{\beta}{2}\|Dx^{k+1}-z\|_{2}^{2}$
$\displaystyle-\frac{\beta}{2}\|Dx^{k}-z^{k+1}\|_{2}^{2}.$

By utilizing Lemma 3.3, we can rewrite equation (60) as follows

$\displaystyle 0\leq$
$\displaystyle\theta(u^{k+1})-\theta(u^{*})+(\omega^{k+1}-\omega^{*})^{T}F(\omega^{*})$
(61) $\displaystyle\leq$
$\displaystyle\frac{1}{2\beta}\|\lambda^{k}-\lambda^{*}\|_{2}^{2}+\frac{\beta}{2}\|Dx^{k}-z^{*}\|_{2}^{2}-\frac{1}{2\beta}\|\lambda^{k+1}-\lambda^{*}\|_{2}^{2}-\frac{\beta}{2}\|Dx^{k+1}-z^{*}\|_{2}^{2}$
$\displaystyle-$
$\displaystyle\frac{1}{2\beta}\|\lambda^{k}-\lambda^{k+1}\|_{2}^{2}-\frac{\beta}{2}\|Dx^{k}-Dx^{k+1}\|_{2}^{2}.$

That is

$\displaystyle\frac{1}{\beta}\|\lambda^{k+1}-\lambda^{*}\|_{2}^{2}+\beta\|Dx^{k+1}-z^{*}\|_{2}^{2}$
(62) $\displaystyle\leq$
$\displaystyle\frac{1}{\beta}\|\lambda^{k}-\lambda^{*}\|_{2}^{2}+\beta\|Dx^{k}-z^{*}\|_{2}^{2}-(\frac{1}{\beta}\|\lambda^{k}-\lambda^{k+1}\|_{2}^{2}+\beta\|Dx^{k}-Dx^{k+1}\|_{2}^{2}).$

Let

$\displaystyle Dx^{*}=z^{*},~{}v=\left(\begin{array}[]{c}\lambda\\\ x\\\
\end{array}\right)~{}\textrm{and}~{}H=\left(\begin{array}[]{cc}\frac{1}{\beta}I_{m}&0\\\
0&\beta D^{T}D\\\ \end{array}\right),$

therefore, the left-hand side of inequality (62) becomes

$\displaystyle\frac{1}{\beta}\|\lambda^{k+1}-\lambda^{*}\|_{2}^{2}+\beta\|Dx^{k+1}-z^{*}\|_{2}^{2}$
(64) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}\lambda^{k+1}-\lambda^{*}\\\
x^{k+1}-x^{*}\\\
\end{array}\right)^{T}\left(\begin{array}[]{cc}\frac{1}{\beta}I_{m}&0\\\
0&\beta D^{T}D\\\
\end{array}\right)\left(\begin{array}[]{c}\lambda^{k+1}-\lambda^{*}\\\
x^{k+1}-x^{*}\\\ \end{array}\right)$ $\displaystyle=$
$\displaystyle(v^{k+1}-v^{*})^{T}\left(\begin{array}[]{cc}\frac{1}{\beta}I_{m}&0\\\
0&\beta D^{T}D\\\ \end{array}\right)(v^{k+1}-v^{*})$ $\displaystyle=$
$\displaystyle\|v^{k+1}-v^{*}\|_{H}^{2}.$

Likewise, the sum of the first two terms on the right-hand side of inequality
(62) is

$\displaystyle\frac{1}{\beta}\|\lambda^{k}-\lambda^{*}\|_{2}^{2}+\beta\|Dx^{k}-Dx^{*}\|_{2}^{2}$
(65) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}\lambda^{k}-\lambda^{*}\\\
x^{k}-x^{*}\\\
\end{array}\right)^{T}\left(\begin{array}[]{cc}\frac{1}{\beta}I_{m}&0\\\
0&\beta D^{T}D\\\
\end{array}\right)\left(\begin{array}[]{c}\lambda^{k}-\lambda^{*}\\\
x^{k}-x^{*}\\\ \end{array}\right)$ $\displaystyle=$
$\displaystyle(v^{k}-v^{*})^{T}\left(\begin{array}[]{cc}\frac{1}{\beta}I_{m}&0\\\
0&\beta D^{T}D\\\ \end{array}\right)(v^{k}-v^{*})$ $\displaystyle=$
$\displaystyle\|v^{k}-v^{*}\|_{H}^{2}$

and the sum of the last two terms on the right-hand side of inequality (62) is

$\displaystyle\frac{1}{\beta}\|\lambda^{k}-\lambda^{k+1}\|_{2}^{2}+\beta\|Dx^{k}-Dx^{k+1}\|_{2}^{2}$
(66) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}\lambda^{k}-\lambda^{k+1}\\\
x^{k}-x^{k+1}\\\
\end{array}\right)^{T}\left(\begin{array}[]{cc}\frac{1}{\beta}I_{m}&0\\\
0&\beta D^{T}D\\\
\end{array}\right)\left(\begin{array}[]{c}\lambda^{k}-\lambda^{k+1}\\\
x^{k}-x^{k+1}\\\ \end{array}\right)$ $\displaystyle=$
$\displaystyle(v^{k}-v^{k+1})^{T}\left(\begin{array}[]{cc}\frac{1}{\beta}I_{m}&0\\\
0&\beta D^{T}D\\\ \end{array}\right)(v^{k}-v^{k+1})$ $\displaystyle=$
$\displaystyle\|v^{k}-v^{k+1}\|_{H}^{2}.$

Since $D\in R^{n\times d}$ is a unit tight frame, we have that $D^{T}D=I_{d}$.
By combining formulas (64), (65), and (66), we complete the proof of the
Theorem 3.4. ∎

According to Theorem 3.4, we know that $H$ is a positive definite matrix, and
inequality (58) implies that the sequence $\\{v^{k}\\}$ is bounded. Assuming
that the initial vector is $v_{0}=(\lambda_{0},x_{0})^{T}$, we can obtain the
following expression by summing both sides of inequality (58)

$\sum^{\infty}_{k=0}\|v^{k}-v^{k+1}\|_{H}^{2}\leq\|v^{0}-v^{*}\|_{H}^{2}.$
(67)

The above equation indicates that
$\lim_{k\rightarrow\infty}\|v^{k}-v^{k+1}\|_{H}^{2}=0$. Therefore, any
subsequence ${v^{k_{j}}}$ of ${v^{k}}$ also has
$\lim_{j\rightarrow\infty}\|v^{k_{j}}-v^{k_{j}+1}\|_{H}^{2}=0$. Suppose there
exists a subsequence that converges to $\bar{v}$, then formula (38) implies
that $\bar{v}$ is the solution of formula (36). This shows that any
accumulation point of the sequence ${v^{k}}$ is a solution of (36). According
to formula (58), ${v^{k}}$ cannot have more than one accumulation point, and
hence ${v^{k}}$ converges to $\bar{v}\in\mathcal{V}^{*}$.

###  3.4 Convergence rate in ergodic sense

Combining with Theorem 3.4, we prove a worst-case $\mathcal{O}(1/t)$
convergence rate in a ergodic sense of the ADMM scheme (36) for cosparse
signal reconstruction problem.

######  Theorem 3.5.

Let the sequence $\\{\omega^{k}\\}$ be generated by (36). Then, for any
positive integer $t$, we have

$\displaystyle\begin{aligned}
&\theta(u^{t})-\theta(u)+(\omega^{t}-\omega)^{T}F(\omega)\\\
\leq&\frac{1}{2(t+1)}[\frac{1}{\beta}\|\lambda^{0}-\lambda\|_{2}^{2}+\beta\|Dx^{0}-z\|_{2}^{2}],~{}\forall\omega\in\Omega\end{aligned}$
(68)

where

$\displaystyle\omega^{t}=\frac{1}{t+1}(\sum_{k=0}^{t}\omega^{k+1}).$ (69)

###### Proof.

For any integer $k$, by (60) we obtain

$\displaystyle\theta(u^{k+1})-\theta(u)+(\omega^{k+1}-\omega)^{T}F(\omega)$
(70) $\displaystyle\leq$
$\displaystyle\frac{1}{2\beta}\|\lambda^{k}-\lambda\|_{2}^{2}+\frac{\beta}{2}\|Dx^{k}-z\|_{2}^{2}-\frac{1}{2\beta}\|\lambda^{k+1}-\lambda\|_{2}^{2}-\frac{\beta}{2}\|Dx^{k+1}-z\|_{2}^{2}.$

Suppose $k=0,1,2,\ldots,t$ are non-negative integers. By summing the left and
right ends of the inequality (70), we deduce that

$\displaystyle\sum_{k=0}^{t}\theta(u^{k+1})-(t+1)\theta(u)+\left[\sum_{k=0}^{t}\omega^{k+1}-(t+1)\omega\right]^{T}F(\omega)$
(71) $\displaystyle\leq$
$\displaystyle\frac{1}{2\beta}\|\lambda^{0}-\lambda\|_{2}^{2}+\frac{\beta}{2}\|Dx^{0}-z\|_{2}^{2},~{}~{}\forall\omega\in\Omega.$

The left and right ends of the inequality (71) are multiplied by
$\frac{1}{t+1}$ at the same time, and let

$\displaystyle\omega^{t}=\frac{1}{t+1}\sum_{k=0}^{t}\omega^{k+1},$ (72)

then, the inequality (71) is equivalent to

$\displaystyle\frac{1}{t+1}\sum_{k=0}^{t}\theta(u^{k+1})-\theta(u)+(\omega^{t}-\omega)^{T}F(\omega)$
(73) $\displaystyle\leq$
$\displaystyle\frac{1}{2(t+1)}[\frac{1}{\beta}\|\lambda^{0}-\lambda\|_{2}^{2}+\beta\|Dx^{0}-z\|_{2}^{2}].$

Given that the function $\theta(u)$ is convex, let

$u^{t}=\frac{1}{t+1}\sum_{k=0}^{t}u^{k+1}=\frac{1}{t+1}(u^{1}+u^{2}+\cdots+u^{t}),$

we can derive the following expression

$\displaystyle\theta(u^{t})=$
$\displaystyle\theta\left[\frac{1}{t+1}(u^{1}+u^{2}+\cdots+u^{t})\right]$ (74)
$\displaystyle\leq$
$\displaystyle\frac{1}{t+1}\left[\theta(u^{1})+\theta(u^{1})+\cdots+\theta(u^{t})\right]$
$\displaystyle=$ $\displaystyle\frac{1}{t+1}\sum_{k=0}^{t}\theta(u^{k+1}).$

By utilizing equations (73) and (74), we complete the proof of the Theorem
3.5. ∎

After $t$-th iterations, then $\omega^{t}$ defined by (72) satisfies

$\displaystyle\begin{aligned}
\tilde{\omega}\in\Omega~{}~{}\textrm{and}~{}\sup_{\omega\in\mathcal{D}_{\tilde{\omega}}}\\{\theta({\tilde{u}})-\theta(u)+(\tilde{\omega}-\omega)^{T}F(\omega)\\}\leq\frac{d}{2t}=\mathcal{O}(\frac{1}{t}),\end{aligned}$

where

$\displaystyle
D_{\tilde{\omega}}=\\{\omega\in\Omega|\|\omega-\tilde{\omega}\|\leq
1\\},~{}~{}d:=\sup\\{\frac{1}{\beta}\|\lambda^{0}-\lambda\|_{2}^{2}+\beta\|Dx^{0}-z\|_{2}^{2}|\omega\in\mathcal{D}_{\tilde{\omega}}\\},$

and $v^{0}=(\lambda^{0},x^{0})$ is the initial iteration point. That means
$\omega^{t}$ is an $\mathcal{O}(\frac{1}{t})$ solution of (37).

##  4 Conclusions

This paper presents a novel approach to analyze the convergence of cosparse
optimization problem. In order to complete the proof of the main theorem, we
first give the overall framework of the ADMM to solve the cosparse
optimization problem, secondly, through three lemmas, this paper gives the
basic inequalities required for the theorem proof, and finally, our analysis
establishes a worst-case convergence rate of $\mathcal{O}(1/t)$, which
demonstrates the effectiveness of our approach.

Researchers currently rely on a range of methods to solve separable convex
optimization problems. Two popular approaches are the generalized symmetric
ADMM and parameterizable proximal point algorithms [33, 34, 35]. These methods
have demonstrated their effectiveness and superiority in various experiments.
In our future work, we plan to explore the potential of combining these
methods to solve the cosparse signal reconstruction problem.

## Funding

This work was supported in part by the National Natural Science Foundation of
China Mathematics Tian Yuan Fund under grant No. 12226323 and 12226315.

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