ISOMAP and machine learning algorithms for the construction of embedded functional connectivity networks of anatomically separated brain regions from resting state fMRI data of patients with Schizophrenia

Ioannis K Gallos; Kostakis Gkiatis; George K Matsopoulos; Constantinos Siettos; Ioannis K Gallos; Kostakis Gkiatis; George K Matsopoulos; Constantinos Siettos

doi:10.3934/Neuroscience.2021016

AIMS Neuroscience

2021, Volume 8, Issue 2: 295-321. doi: 10.3934/Neuroscience.2021016

Previous Article Next Article

Research article Topical Sections

ISOMAP and machine learning algorithms for the construction of embedded functional connectivity networks of anatomically separated brain regions from resting state fMRI data of patients with Schizophrenia

1.
School of Applied Mathematical and Physical Sciences, National Technical University of Athens, Greece
2.
School of Electrical and Computer Engineering, National Technical University of Athens, Greece
3.
Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”, Università degli Studi di Napoli Federico II, Italy

Received: 01 February 2021 Accepted: 18 February 2021 Published: 19 February 2021

We construct Functional Connectivity Networks (FCN) from resting state fMRI (rsfMRI) recordings towards the classification of brain activity between healthy and schizophrenic subjects using a publicly available dataset (the COBRE dataset) of 145 subjects (74 healthy controls and 71 schizophrenic subjects). First, we match the anatomy of the brain of each individual to the Desikan-Killiany brain atlas. Then, we use the conventional approach of correlating the parcellated time series to construct FCN and ISOMAP, a nonlinear manifold learning algorithm to produce low-dimensional embeddings of the correlation matrices. For the classification analysis, we computed five key local graph-theoretic measures of the FCN and used the LASSO and Random Forest (RF) algorithms for feature selection. For the classification we used standard linear Support Vector Machines. The classification performance is tested by a double cross-validation scheme (consisting of an outer and an inner loop of “Leave one out” cross-validation (LOOCV)). The standard cross-correlation methodology produced a classification rate of 73.1%, while ISOMAP resulted in 79.3%, thus providing a simpler model with a smaller number of features as chosen from LASSO and RF, namely the participation coefficient of the right thalamus and the strength of the right lingual gyrus.

Keywords:

Citation: Ioannis K Gallos, Kostakis Gkiatis, George K Matsopoulos, Constantinos Siettos. ISOMAP and machine learning algorithms for the construction of embedded functional connectivity networks of anatomically separated brain regions from resting state fMRI data of patients with Schizophrenia[J]. AIMS Neuroscience, 2021, 8(2): 295-321. doi: 10.3934/Neuroscience.2021016

Related Papers:

[1]	Tolga Zaman, Cem Kadilar . Exponential ratio and product type estimators of the mean in stratified two-phase sampling. AIMS Mathematics, 2021, 6(5): 4265-4279. doi: 10.3934/math.2021252
[2]	Khazan Sher, Muhammad Ameeq, Sidra Naz, Basem A. Alkhaleel, Muhammad Muneeb Hassan, Olayan Albalawi . Developing and evaluating efficient estimators for finite population mean in two-phase sampling. AIMS Mathematics, 2025, 10(4): 8907-8925. doi: 10.3934/math.2025408
[3]	Sohaib Ahmad, Sardar Hussain, Muhammad Aamir, Faridoon Khan, Mohammed N Alshahrani, Mohammed Alqawba . Estimation of finite population mean using dual auxiliary variable for non-response using simple random sampling. AIMS Mathematics, 2022, 7(3): 4592-4613. doi: 10.3934/math.2022256
[4]	Yasir Hassan, Muhammad Ismai, Will Murray, Muhammad Qaiser Shahbaz . Efficient estimation combining exponential and ln functions under two phase sampling. AIMS Mathematics, 2020, 5(6): 7605-7623. doi: 10.3934/math.2020486
[5]	Saman Hanif Shahbaz, Aisha Fayomi, Muhammad Qaiser Shahbaz . Estimation of the general population parameter in single- and two-phase sampling. AIMS Mathematics, 2023, 8(7): 14951-14977. doi: 10.3934/math.2023763
[6]	Amber Yousaf Dar, Nadia Saeed, Moustafa Omar Ahmed Abu-Shawiesh, Saman Hanif Shahbaz, Muhammad Qaiser Shahbaz . A new class of ratio type estimators in single- and two-phase sampling. AIMS Mathematics, 2022, 7(8): 14208-14226. doi: 10.3934/math.2022783
[7]	Sanaa Al-Marzouki, Christophe Chesneau, Sohail Akhtar, Jamal Abdul Nasir, Sohaib Ahmad, Sardar Hussain, Farrukh Jamal, Mohammed Elgarhy, M. El-Morshedy . Estimation of finite population mean under PPS in presence of maximum and minimum values. AIMS Mathematics, 2021, 6(5): 5397-5409. doi: 10.3934/math.2021318
[8]	Sohaib Ahmad, Sardar Hussain, Javid Shabbir, Muhammad Aamir, M. El-Morshedy, Zubair Ahmad, Sharifah Alrajhi . Improved generalized class of estimators in estimating the finite population mean using two auxiliary variables under two-stage sampling. AIMS Mathematics, 2022, 7(6): 10609-10624. doi: 10.3934/math.2022592
[9]	Khazan Sher, Muhammad Ameeq, Muhammad Muneeb Hassan, Basem A. Alkhaleel, Sidra Naz, Olyan Albalawi . Novel efficient estimators of finite population mean in stratified random sampling with application. AIMS Mathematics, 2025, 10(3): 5495-5531. doi: 10.3934/math.2025254
[10]	Hleil Alrweili, Fatimah A. Almulhim . Estimation of the finite population mean using extreme values and ranks of the auxiliary variable in two-phase sampling. AIMS Mathematics, 2025, 10(4): 8794-8817. doi: 10.3934/math.2025403

Abstract

1. Introduction

1.1. Introduction and notation in simple random sampling

It is a well-known fact, that at large scale survey sampling, the use of several auxiliary variables improve the precision of the estimators. In survey sampling, researchers have already attempted to obtain the estimates for population parameter such as mean, median etc, that posses maximum statistical properties. For that purpose a representative part of population is needed, when population of interest is homogeneous then one can use simple random sampling (SRS) for selecting units. In some situations, information available in the form of attributes, which is positively correlated with study variables. Several authors including Naik and Gupta ^[1], Jhajj ^[2], Abd-Elfattah ^[3], Koyuncu ^[4], Solanki ^[5], Sharma ^[6] and Malik ^[7] proposed a set of estimators, taking the advantages of bi-serial correlation between auxiliary and study variables, utilizing information on single auxiliary attribute. Verma ^[8], Malik ^[7], Solanki et al., ^[9] and Sharma ^[10] suggested some estimators utilizing information on two auxiliary attributes in $\text{SRS}$ , Mahdizadeh and Zamanzade ^[11] developed a kernel-based estimation of $P(X > Y)$ in ranked-set sampling, SinghPal and Solanki ^[12] developed a new class of estimators of finite population mean survey sampling and Mahdizadeh and Zamanzade ^[13] suggest a smooth estimation of a reliability function in ranked set sampling, further more Hussain et al., ^[14] and Al-Marzouki et al., ^[15] also work in this side.

In this article, we consider the problem of estimating the finite population mean using the auxiliary proportion under simple and two phase sampling scheme. The mathematical expression of the bias and mean squared error of the proposed estimator are derived under first order of approximation. The performance of proposed class of estimator is compared with that of the existing estimators both theoretically and numerically. In terms of percentage relative efficiency (PRE), it is found that proposed class of estimator outperforms the existing ones.

Let $U = \{u{_1}, u{_2}, ..., u{_N}\}$ represent a finite population of size N distinct units, assumed that a sample of size n units is drawn from this population U using simple random sampling without replacement. Let $y_{i}$ and $\phi{_i}_j$ (i = 1, 2) denotes the observations on variable $y$ and $\phi_{i}$ (i = 1, 2) for the $j^{th}$ unit (j = 1, 2, ..., N).

$\phi_{ij} = 1$ , if $i^{th}$ unit posses atrributes

$\phi_{ij} = 0$ , $\text{otherwise}$

$P{_j} = \sum_{i}^{N}\phi{_i}_j = A{_j}/N, (j = 1, 2)$ and $p{_j} = \sum_{i}^{N}\phi{_i}_j = a{_j}/n, (j = 1, 2)$ are the population and sample proportions of auxiliary variable respectively. Let $\bar{Y} = \frac{\sum_{i = 1}^{N}y_{i}}{N}$ , $\bar{y} = \frac{\sum_{i = 1}^{n}y_{i}}{n}$ be the population and sample mean of the study variable $y$ . $S^{2}_{\phi_{j}y} = \sum_{i = 1}^{N}\frac{(\phi_{ij}-P_{j})(y_{i}-\bar{_Y})}{N-1}, (j = 1, 2)$ are the variations between the study and the auxiliary attributes. $S^{2}_{\phi_{1} \phi_{2}} = \sum_{i = 1}^{N}\frac{(\phi_{i1}-P_{1})(\phi_{i2}-P_{2})}{N-1}$ are the variations between the auxiliary attributes. $\rho_{y \phi_{j}} = \frac{S_{\phi_{j} y}}{S_{y}S_{\phi}}$ represents the point bi-serial correlation between the study variable $y$ and the two auxiliary attributes $p_{1}$ and $p_{2}$ respectively. $\rho_{\phi_{1} \phi_{2}} = \frac{S_{\phi_{1} \phi{2}}}{S_{\phi_{1}}S_{\phi_{2}}}$ represents the point bi-serial correlation between the two auxiliary attributes $p_{1}$ and $p_{2}$ respectively.

Let us define, $e_{0} = \frac{\bar{y}-\bar{Y}}{\bar{Y}}$ , $e_{1} = \frac{p_{1}-P_{1}}{P_{1}}$ , $e_{2} = \frac{p_{2}-P_{2}}{P_{2}}$ ,

such that, $E(e_{i}) = 0$ $(i = 0, 1, 2)$ ,

$E(e^{2}_{0}) = fC^{2}_{y} = V_{200}$ , $E(e^{2}_{1}) = fC^{2}_{\phi_{1}^2} = V_{020}$ , $E(e^{2}_{2}) = fC^{2}_{\phi_{2}^2} = V_{002}$ ,

$E(e_{0}e_{1}) = f\rho_{y\phi_{1}}C_{y}C_{\phi_{1}} = V_{110}$ , $E(e_{0}e_{2}) = f\rho_{y\phi_{2}}C_{y}C_{\phi_{2}} = V_{101}$ ,

$E(e_{1}e_{2}) = f\rho_{\phi_{1}\phi_{2}}C_{\phi_{1}}C_{\phi_{2}} = V_{011}$ .

Where $C_{{y}} = \frac{S_{y}}{\bar{Y}}$ , $C_{\phi_{j}} = \frac{S_{\phi_{j}}}{P_{j}}, (j = 1, 2),$ is the co-efficient of variation of the study and auxiliary attribute. $S^{2}_{y} = \sum_{i = 1}^{N}\frac{(y_{i}-\bar{Y})^2}{N-1}$ , $S^{2}_{\phi_{j}} = \sum_{i = 1}^{N}\frac{(\phi_{ij}-P_{j})^2}{N-1}, (j = 1, 2),$ is the variance of study and auxiliary attribute. $f = \left(\frac{1}{n}-\frac{1}{N}\right)$ is the correction factor.

The rest of the paper is organized as follows. In Sections 1.1 and 1.2, introduction and notations are given for simple random sampling and two phase sampling. In Sections 2.1 and 2.3, we discussed some existing estimators of the finite population mean for both sampling designs. The proposed estimators are given in Sections 2.2 and 2.4. In Sections 3.1 and 3.2, theoretical comparisons are conducted. While in Sections 4.1 and 4.2 we focus on empirical studies. Finally, application and conclusions are drawn in Sections 5 and 6.

1.2. Introduction and notation in two phase sampling

The precision of estimate can be increased by using two methodologies. Firstly the precision may be increased by using using adequate sampling design for the estimated variable. Secondly the precision may be increased by using an appropriate estimation procedure, i.e. some auxiliary information which is closely associated with the variable under study. In application there exist a situation when complete auxiliary information or attribute is not available or information on that attribute is expensive. In that case, a method of two phase sampling or double sampling is used to obtain the estimates of unknown population parameters. In two phase sampling, a large preliminary sample $(n^\prime)$ is selected by $\text{SRSWOR}$ to obtain the estimate of unknown parameter of the auxiliary variable at first phase and the information on the auxiliary variable is collected, which is use to estimate the unknown auxiliary variable. Then a sub sample $(n < n^\prime)$ is selected at second phase and both the study and auxiliary variables are collected. Here we assume that Population proportion $(P_{1})$ is unknown and introduce an improved estimator to estimate the population mean. Kiregyera ^[16], Mohanty ^[17], Malik ^[7] and Haq ^[18] used two auxiliary variables in two phase sampling for the better estimation of mean.

An example in this context is while estimating the yield of a crop, it is likely that the area under the crop may be unknown but the area of each farm may be known. Then $y$ , $P_{1}$ and $P_{2}$ respectively are the yield area under the crop and area under cultivation.

Consider a finite population $U = (u_{1}, u_{2}...u_{N})$ of size N and let $y_{i}$ , $\phi_{i1}$ and $\phi_{i2}$ is the information on the study variable and two auxiliary attributes associated with each unit $u_{i}(i = 1, 2, ..., N)$ of the population such that:

$\phi{_i}_j = 1$ , if the $i^{th}$ unit in the population possesses auxiliary attribute $\phi{_j}$ , $\phi{_i}_j = 0$ otherwise.

We assume that the population mean of the first auxiliary proportion $P_{1}$ is unknown but the same information is known for the second proportion. Let $p{_j}^\prime = \frac{\sum_{i}^{n^\prime}\phi{_i}_j}{n} = a{_j}/n^\prime$ for $j = 1, 2$ be the estimate of $P_{j}$ obtained from the first phase sample of size $n^\prime$ , drawn by using $\text{SRSWOR}$ from the population of N units. Let $\bar{y} = \sum_{i}^{n}y_{i}$ and $p{_1} = \frac{\sum_{i}^{n}\phi{_i}_1}{n} = a{_1}/n$ be the estimates of $\bar{Y}$ and $P_{1}$ respectively, obtained from a second sample of size n, drawn from the first phase $n^\prime$ using $\text{SRSWOR}$ .

To obtain the bias and $\text{MSE}$ for estimators in two phase sampling we define the error terms as follows:

$e_{0} = \frac{\bar{y}-\bar{Y}}{\bar{Y}}$ , $e_{1} = \frac{p_{1}-P_{1}}{P_{1}}$ , $e_{2^\prime} = \frac{p_{2}^\prime-P_{2}}{P_{2}}$ , $e_{1}^\prime = \frac{p_{1}^\prime-P_{1}}{P_{1}}$ .

such that:

$E(e_{0}) = E(e_{1}) = E(e_{2}^\prime) = E(e_{1}^\prime) = 0$ ,

$E(e^{2}_{0}) = fC^{2}_{y} = V_{200}$ , $E(e^{2}_{1}) = fC^{2}_{\phi_{1}} = V_{020}$ $E(e^{2}_{2}) = fC^{2}_{\phi_{2}} = V_{002},$

$E(e_{0}e_{1}) = f\rho_{y\phi_{1}}C_{y}C_{\phi_{1}} = V_{110},$ $E(e_{0}e_{2}) = f\rho_{y\phi_{2}}C_{y}C_{\phi_{2}} = V_{101}$ ,

$E(e_{1}e_{2}) = f\rho_{\phi_{1}\phi_{2}}C_{\phi_{1}}C_{\phi_{2}} = V_{011},$ $E(e_{1}^\prime e_{0}) = f^\prime\rho_{y\phi_{1}}C_{y}C_{\phi_{1}} = V_{110^\prime}$ ,

$E(e_{0}^\prime e_{2}) = f^\prime\rho_{y\phi_{2}}C_{y}C_{\phi_{2}} = V_{101}^\prime$ , $E(e_{1}^2\prime) = f^\prime C^{2}_{\phi_{1}} = V_{020}^\prime$ ,

$E(e^{2\prime}_{2}) = f^\prime C^{2}_{\phi_{2}^2} = V_{002}^\prime$ , $f = \frac{1}{n}-\frac{1}{N}$ , $f^\prime = \frac{1}{n^\prime}-\frac{1}{N},$

$\bar{Y} = N\sum_{i = 1}^{N}y_{i}$ , $\bar{y} = n\sum_{i = 1}^{n}y_{i},$

$S^{2}_{y} = \sum_{i = 1}^{N}\frac{(y_{i}-\bar{Y})^2}{N-1}$ , $S^{2}_{y\phi_{j}} = \sum_{i = 1}^{N}\frac{(\phi_{ij}-P_{j})(y_{i}-\bar{_Y})}{N-1},$

$S^{2}_{\phi_{1} \phi_{2}} = \sum_{i = 1}^{N}\frac{(\phi_{i1}-P_{1})(\phi_{i2}-P_{2})}{N-1}$ , $C_{{y}} = \frac{S_{y}}{\bar{Y}},$

$C_{\phi_{j}} = \frac{S_{\phi_{j}}}{P_{j}},$ $S^{2}_{\phi_{j}} = \sum_{i = 1}^{N}\frac{(\phi_{ij}-P_{j})^2}{N-1},$

$s^{2}_{\phi_{j}} = \sum_{i = 1}^{n}\frac{(\phi_{ij}-p_{j})^2}{n-1},$ represents the sample variance of size n,

$s^{2}_{\phi_{j^\prime}} = \sum_{i = 1}^{n^\prime}\frac{(\phi_{ij}-p_{j})^2}{n^\prime-1},$ represents the sample variance of size $n^\prime$

$\rho_{y \phi_{j}} = \frac{S_{\phi_{j} y}}{S_{y}S_{\phi}}$ represent point bi-serial correlation between the study variable $(y)$ and the two auxiliary attributes $(P_{1})$ and $(p_{2})$ .

$\rho_{\phi_{1} \phi_{2}} = \frac{S_{\phi_{1} \phi{2}}}{S_{\phi_{1}}S_{\phi_{2}}}$ represent point bi-serial correlation between the two auxiliary attributes $(P_{1})$ and $(P_{2})$ respectively.

2. Existing and proposed estimators

2.1. Existing estimators in simple random sampling

In order to have an estimate of the study variable, using information of population proportion $P$ , Naik ^[1] proposed the following estimators respectively.

$\begin{equation} t_{U} = \bar{y}. \end{equation}$

(2.1)

The $\text{MSE}$ of $t_{U}$ is given by

$\begin{equation} \text{MSE}({t_{U}}) = \bar{Y}^2V_{200}. \end{equation}$

(2.2)

Naik ^[1] the following estimator respectively

$\begin{equation} t{_A} = \bar{y}\left(\frac{P_1}{p_1}\right), \end{equation}$

(2.3)

$\begin{equation} t{_B} = \bar{y}\left(\frac{p_2}{P_2}\right), \end{equation}$

(2.4)

$\begin{equation} t{_C} = \bar{y}\, \, exp\left( \frac{{P_1}-p{_1}}{{P_1}+{p_1}}\right) , \end{equation}$

(2.5)

$\begin{equation} t{_D} = \bar{y}\, \, exp\left( \frac{{p_2}-P{_2}}{{p_2}+{P_2}}\right) . \end{equation}$

(2.6)

The $\text{MSE}$ expressions of the estimators $t{_A}$ , $t{_B}$ , $t{_C}$ and $t{_D}$ are respectively given as

$\begin{equation} \text{MSE}(t{_A})\cong\bar{Y}^2(V_{200}-2V_{110}+V_{020}), \end{equation}$

(2.7)

$\begin{equation} \text{MSE}(t{_B})\cong\bar{Y}^2(V_{200}+2V_{101}+V_{002}), \end{equation}$

(2.8)

$\begin{equation} \text{MSE}(t{_C})\cong\bar{Y}^2(V_{200}-V_{110}+\frac{1}{4}V_{020}), \end{equation}$

(2.9)

$\begin{equation} \text{MSE}(t{_D})\cong\bar{Y}^2(V_{200}+V_{101}+\frac{1}{4}V_{002}). \end{equation}$

(2.10)

Malik ^[7] proposed exponential type estimator as

$\begin{equation} t_{MS} = \bar{y}\, \, exp\left( \frac{{P_1}-p{_1}}{{P_1}+{p_1}}\right)^{\gamma_1}exp\left( \frac{{P_2}-p{_2}}{{P_2}+{p_2}}\right)^{\gamma_2} +b_{1}(P_{1}-p_{1})+b_{2}(P_{2}-p_{2}), \end{equation}$

(2.11)

where $b_{1} = \frac{s_{y\phi_{1}}}{s_{\phi_{1}}^2}$ and $b_{2} = \frac{s_{y\phi_{2}}}{s_{\phi_{2}}^2}$ are the sample regression coefficients. $\gamma_{1}$ and $\gamma_{2}$ are two unknown constants. The optimum values of these constants are given as:

$\begin{equation*} \gamma_{1(opt)} = \frac{2\left\lbrace -P_{1}\beta_{1}C_{\phi_{1}}(-1+\rho_{\phi_{1}\phi_{2}}^2)+\bar{Y}C_{y}(-\rho_{y\phi_{1}}+\rho_{\phi_{1}\phi_{2}}\rho_{y\phi_{2}})\right\rbrace }{\bar{Y}C_{\phi_{1}}(-1+\rho_{\phi_{1}\phi_{2}}^2)}, \end{equation*}$

$\begin{equation*} \gamma_{2(opt)} = \frac{2\left\lbrace -P_{1}\beta_{2}C_{\phi_{1}}(-1+\rho_{\phi_{1}\phi_{2}}^2)+\bar{Y}C_{y}(-\rho_{y\phi_{2}}+\rho_{\phi_{1}\phi_{2}}\rho_{y\phi_{1}})\right\rbrace }{\bar{Y}C_{\phi_{2}}(-1+\rho_{\phi_{1}\phi_{2}}^2)}. \end{equation*}$

where $\beta_{1} = \frac{S_{y\phi_{1}}}{S_{\phi_{1}}^2}$ and $\beta_{2} = \frac{S_{y\phi_{2}}}{S_{\phi_{2}}^2}$ , are the regression coefficients. The minimum mean squared error for the optimum values of $\gamma_{1}$ and $\gamma_{2}$ are given as:

$\begin{equation} \text{MSE}(t_{MS min})\cong f\bar{Y}^2C_{y}^2(1-R_{y\phi_{1}\phi_{2}}^2), \end{equation}$

(2.12)

where $R = \frac{\rho_{\phi_{1}y}^2+\rho_{\phi_{2}y}^2-2\rho_{\phi_{1}y}\rho_{\phi_{2}y}\rho_{\phi_{1}\phi_{2}}}{1-\rho_{\phi_{1}\phi_{2}}^2 }$ is the multiple correlation of y on $\phi_{1}$ and $\phi_{2}$ .

We used some formulas for readers to easily understand and pick-out the difficulty of long equations.

2.2. The proposed class of estimators in simple random sampling

We proposed generalized class of estimators for estimating mean in simple random sampling using two auxiliary attributes, as

$\begin{equation} t_{RPR} = k_{1} \bar{y}-k_{2}(p_{1}-P_{1}) \bigg[\alpha\left\lbrace 2-exp\left( \frac{\eta(p_{2}-P_{2})}{\eta(p_{2}+P_{2})+2\lambda}\right) \right\rbrace +(1-\alpha)exp\left( \frac{ \eta (P_{2}-p_{2})}{\eta(P_{2}+p_{2})+2\lambda}\right) \bigg], \end{equation}$

(2.13)

where $k_{1}$ and $k_{2}$ are suitable constants whose values are to be determined such that $\text{MSE}$ of $t_{RPR}$ is minimum; $\eta$ and $\lambda$ are either real numbers or functions of known parameters of the auxiliary attribute $\phi_{2}$ such as coefficient of variation $(C_{\phi_{2}})$ , coefficient of kurtosis $(\beta_{\phi_{2}})$ and $\alpha$ is the scalar $(0\leq\alpha\leq1)$ for designing different estimators. Let $\bar{Y}$ and $(P_{1}, P_{2})$ be the population means of the study variable and auxiliary proportions respectively. $\bar{y}$ and $(p_{1}, p_{2})$ be the sample means of the study variable and auxiliary proportions respectively.

Putting $\alpha = 1$ and $\alpha = 0$ in (2.13), we get the following estimators.

For $\alpha = 1$ , the suggested class of estimators reduces to:

$\begin{equation*} t_{RPR(\alpha = 1)} = k_{1}\bar{y}-k_{2}( p_{1}-P_{1}) \left[2- exp \left\lbrace \frac{\eta(p_{2}-P_{2})}{\eta( p_{2}-P_{2})+2\lambda} \right\rbrace\right]. \end{equation*}$

For $\alpha = 0$ , the suggested class of estimators reduces to

$\begin{equation*} t_{RPR(\alpha = 0)} = k_{1}\bar{y}-k_{2}( p_{1}-P_{1}) \left[ exp \left\lbrace \frac{\eta(P_{2}-p_{2})}{\eta( P_{2}-p_{2})+2\lambda} \right\rbrace\right]. \end{equation*}$

A set of of new estimators generated from Eq (2.13) using suitable use of $\alpha$ , $\eta$ and $\lambda$ are listed in Table 1.

Table 1. Set of estimators generated from estimator

$t_{RPR}$ .

$\text{Subset of proposed estimator}$	$\alpha$	$\eta$	$\lambda$
$t_{RPR1}=k_{1}\bar{y}-k_{2}(p_{1}-P_{1}) \left[exp \left\lbrace \frac{C_{\phi_{2}}(P_{2}-p_{2})}{C_{\phi_{2}}(P_{2}-p_{2})+2\beta_{2}\phi_{2}} \right\rbrace\right]$	0	$C_{\phi_{2}}$	$\beta_{2\phi_{2}}$
$t_{RPR2}=k_{1}\bar{y}-k_{2}(p_{1}-P_{1}) \left[exp\left\lbrace \frac{P_{2}(P_{2}-p_{2})}{P_{2}(P_{2}-p_{2})+2} \right\rbrace\right]$	0	$P_{2}$	$1$
$t_{RPR3}=k_{1}\bar{y}-k_{2}(p_{1}-P_{1}) \left[exp \left\lbrace \frac{(P_{2}-p_{2})}{(P_{2}-p_{2})+2C_{\phi_{2}}} \right\rbrace\right]$	0	1	$C_{\phi_{2}}$
$t_{RPR4}=k_{1}\bar{y}-k_{2}(p_{1}-P_{1})\left[exp \left\lbrace \frac{(P_{2}-p_{2})}{(P_{2}-p_{2})+2} \right\rbrace\right]$	0	1	1
$t_{RPR5}=k_{1}\bar{y}-k_{2}(p_{1}-P_{1})\left[2-exp\left\lbrace \frac{C_{\phi_{2}}(p_{2}-P_{2})}{C_{\phi_{2}}(p_{2}-P_{2})+2\beta_{2}\phi_{2}} \right\rbrace\right]$	1	$C_{\phi_{2}}$	$\beta_{2\phi_{2}}$
$t_{RPR6}=k_{1}\bar{y}-k_{2}(p_{1}-P_{1})\left[2- exp\left\lbrace \frac{P_{2}(p_{2}-P_{2})}{P_{2}(p_{2}-P_{2})+2} \right\rbrace\right]$	1	$P_{2}$	$1$
$t_{RPR7}=k_{1}\bar{y}-k_{2}(p_{1}-P_{1})\left[2-exp\left\lbrace\frac{(p_{2}-P_{2})}{(p_{2}-P_{2})+2C_{\phi_{2}}}\right\rbrace\right]$	1	1	$C_{\phi_{2}}$
$t_{RPR8}=k_{1}\bar{y}-k_{2}(p_{1}-P_{1})\left[2-exp\left\lbrace \frac{(p_{2}-P_{2})}{(p_{2}-P_{2})+2}\right\rbrace\right]$	1	1	1

| Show Table

DownLoad: CSV

Expressing Eq (2.13) in terms of e's we have

$\begin{equation} t_{RPR} = k_{1}\bar{Y}(1+e_{0})-k_{2}P_{1}e_{1}\bigg[\alpha\left\lbrace 2-\left(1-\gamma e_{2}-\frac{1}{2} \gamma^2 e^{2}_{2}\right)\right\rbrace+ \left( 1-\alpha\right)\left( 1-\gamma e_{2}+\frac{3}{2}\gamma^2e^{2}_{2}\right)\bigg], \end{equation}$

(2.14)

where $\gamma = \frac{\eta P_{2}}{2(\eta P_{2}+\lambda)}.$

To the first degree of approximation, we have:

$\begin{equation} t_{RPR}-\bar{Y}\cong k_{1} \bar{Y}+k_{1} \bar{Y}e_{0}-k_{2}P_{1}e_{1}-k_{1}\gamma \bar{Y}e_{2}-\gamma \bar{Y} k_{1}e_{2}e_{0}+\frac{\bar{Y}k_{1}e^2_{2}\gamma^2 3}{2}-\alpha \bar{Y} k_{1}e^{2}_{2} \gamma^2+\gamma k_{2}P_{1} e_{1}e_{2}-\bar{Y}. \end{equation}$

(2.15)

Taking expectation of the above equation we get bias of $t_{RPR}$ , given by:

$\begin{equation} Bias(t_{RPR})\cong k_{1}\bar{Y}-\gamma\bar{Y}k_{1}V_{101}+\bar{Y}k_{1}V_{002}\gamma^2(\frac{3}{2}-\alpha)+\gamma k_{2}P_{1}V_{011}. \end{equation}$

(2.16)

Squaring both sides of Eq (2.15) and taking expectations of both sides, we get the $\text{MSE}$ of the estimator $t_{RPR}$ to the first order of approximation, as

$\begin{equation*} E(t_{RPR}-\bar{Y})^2\cong\bar{Y}^2+\bar{Y}^2k_{1}^2\left( 1-4\gamma V_{101}-2\alpha V_{002}\gamma^2+4\gamma^2V_{002}+V_{200}\right) \end{equation*}$

$\begin{equation*} \qquad\qquad-k_{1}\bar{Y}^2\left( 2-2\alpha V_{002}\gamma^2-2\gamma V_{101}+3V_{002}\gamma^2\right) \end{equation*}$

$\begin{equation*} +2k_{1}k_{2}\bar{Y}\left( 2\gamma P_{1}V_{011}-P_{1}V_{110}\right) \end{equation*}$

$\begin{equation} -2k_{2}\bar{Y}\left( \gamma P_{1}V_{011}\right) +k_{2}^2\left( P_{1}^2V_{020}\right), \end{equation}$

(2.17)

$\begin{equation} \text{MSE}t_{RPR}\cong\bar{Y}^2+\bar{Y}^2k_{1}^2A-k_{1}\bar{Y}^2B+2k_{1}k_{2}\bar{Y}C-2k_{2}\bar{Y}D+k_{2}^2E. \end{equation}$

(2.18)

where

$\begin{equation*} A = 1-4\gamma V_{101}-2\alpha V_{002} \gamma^2+4\gamma^2V_{002}+V_{200}, \end{equation*}$

$\begin{equation*} B = 2-2\alpha V_{002}\gamma^2-2\gamma V_{101}+3V_{002}\gamma^2, \end{equation*}$

$\begin{equation*} C = 2\gamma^2P_{1}V_{011}-P_{1}V_{110}, \quad D = \gamma P_{1}V_{011}, \quad E = P_{1}^2V_{020}. \end{equation*}$

The optimum values of $k_{1}$ and $k_{2}$ are obtained by minimizing Eq (2.18) and is given by

$\begin{equation*} k_{1} = \frac{BE-2CD}{2\left( AE-C^2\right)}, \end{equation*}$

and

$\begin{equation*} k_{2} = \frac{\bar{Y}\left( 2AD-BC\right)}{2\left( AE-C^2\right)}, \end{equation*}$

Substituting the optimum values of $k_{1}$ and $k_{2}$ in Eq $(2.18)$ we get the minimum $\text{MSE}$ of $t_{RPR}$ as:

$\begin{equation} \text{MSE}(t_{(RPR) min}) = \frac{\bar{Y}^2\left( 1-4AD^2+B^2E-4BCD\right)}{4\left(AE-C^2\right)}. \end{equation}$

(2.19)

The minimum MSE of the proposed estimator tRPR at Eq (2.19) depends upon many parametric constants, we use these constant for readers to easily understand and for notation convenient.

2.3. Existing estimators in two phase sampling

The usual mean per unit estimator in two phase sampling is:

$\begin{equation} t_{U}^\prime = \bar{y}. \end{equation}$

(2.20)

The $\text{MSE}$ of ${t_{U}^\prime}$ is given by

$\begin{equation} \text{MSE}(t_{U}^\prime) = \bar{Y}^2V_{200}. \end{equation}$

(2.21)

The Naik ^[1] estimators in two phase sampling are :

$\begin{eqnarray} t{_A}^\prime = \bar{y}\left(\frac{p_1^\prime}{p_1}\right), \end{eqnarray}$

(2.22)

$\begin{eqnarray} t{_B}^\prime = \bar{y}\left(\frac{P_2}{p_2^\prime}\right), \end{eqnarray}$

(2.23)

$\begin{eqnarray} t{_C}^\prime = \bar{y}\, \, exp\left( \frac{{p_1}^\prime-p{_1}}{{p_1}^\prime+{p_1}}\right), \end{eqnarray}$

(2.24)

$\begin{eqnarray} t{_D}^\prime = \bar{y}\, \, exp\left( \frac{{P_2}-p{_2}^\prime}{{P_2}+{p_2}^\prime}\right). \end{eqnarray}$

(2.25)

The $\text{MSE}$ expressions of estimators $t{_A}^\prime$ , $t{_B}^\prime$ , $t{_C}^\prime$ and $t{_D}^\prime$ are respectively given as:

$\begin{equation} \text{MSE}(t{_A}^\prime)\cong\bar{Y}^2\left( V_{200}+V_{020}-V_{020}^\prime+2V_{110}^\prime-2V_{110}^\prime\right) , \end{equation}$

(2.26)

$\begin{equation} \text{MSE}(t{_B}^\prime)\cong\bar{Y}^2\left(V_{200}+V_{002}^\prime+2V_{101}^\prime\right) , \end{equation}$

(2.27)

$\begin{equation} \text{MSE}(t{_C}^\prime)\cong\bar{Y}^2\left(V_{200}+V_{110}^\prime-V_{110}-\frac{1}{4}V_{020}^\prime+\frac{1}{4}V_{020}\right), \end{equation}$

(2.28)

$\begin{equation} \text{MSE}(t{_D}^\prime)\cong\bar{Y}^2\left(V_{200}+\frac{1}{4}V_{002}^\prime+V_{101}^\prime\right). \end{equation}$

(2.29)

Malik ^[7] used exponential type estimator with regression coefficients in two phase sampling which is given by:

$\begin{equation} t_{MS} = \bar{y}\, \, exp\left( \frac{{p_1}^\prime-p{_1}}{{p_1}^\prime+{p_1}}\right)^{\delta_1} exp\left( \frac{{P_2}-p{_2}^\prime}{{P_2}+{p_2}^\prime}\right)^{\delta_2} +b_{1}(P_{1}-p_{1})+b_{2}(P_{2}-p_{2}), \end{equation}$

(2.30)

where $b_{1} = \frac{s_{y\phi_{1}}}{s_{\phi_{1}}^2}$ and $b_{2} = \frac{s_{y\phi_{2}}}{s_{\phi_{2}}^2}$ are the sample regression coefficients. $\delta_{1}$ and $\delta_{2}$ are two unknown constants. The optimum values of these constants are given as:

$\begin{equation*} \delta_{1(opt)} = \frac{-2P_{1}\beta_{1}}{\bar{Y}} + \frac{2C_{y}\rho_{y\phi_{1}}}{C_{\phi_{1}}} , \end{equation*}$

$\begin{equation*} \delta_{2(opt)} = \frac{-2P_{2}\beta_{2}}{\bar{Y}} + \frac{2C_{y}\rho_{y\phi_{2}}}{C_{\phi_{2}}} , \end{equation*}$

where, $\beta_{1} = \frac{S_{y\phi_{1}}}{S_{\phi_{1}}^2}$ and $\beta_{2} = \frac{S_{y\phi_{2}}}{S_{\phi_{2}}^2}$ are the regression coefficients.

The minimum mean square error for the optimum values of $\delta_{1}$ and $\delta_{2}$ are given as:

$\begin{equation} \text{MSE}(t_{MS min})\cong f\bar{Y}^2C_{y}^2\left\lbrace -f(-1+\rho_{y\phi_{1}}^2)+ \lambda(\rho_{y\phi_{1}}^2-\rho_{y\phi_{2}}^2)\right\rbrace. \end{equation}$

(2.31)

2.4. A generalized proposed class of estimators in two phase sampling

We suggest a generalized exponential estimator when $P_{1}$ is unknown and $P_{2}$ is known:

$\begin{equation} t_{RPR}^\prime = k_{1}^\prime \bar{y}-k_{2}^\prime(p_{1}-p_{1}^\prime) \bigg[\alpha^\prime\left\lbrace 2-exp\left( \frac{\eta^\prime(p_{2}^\prime-P_{2})}{\eta^\prime(p_{2}^\prime+P_{2})+2\lambda^\prime}\right) \right\rbrace +(1-\alpha^\prime)exp\left( \frac{ \eta^\prime (P_{2}-p_{2}^\prime)}{\eta^\prime(P_{2}+p_{2}^\prime)+2\lambda^\prime}\right)\bigg ]. \end{equation}$

(2.32)

where $k_{1}^\prime$ and $k_{2}^\prime$ are suitable constants whose value are to be determined such that $\text{MSE}$ of $t_{RPR}$ is minimum. $\eta^\prime$ and $\lambda^\prime$ are either real numbers or functions of known parameters of the auxiliary attribute $\phi_{2}$ such as coefficient of variation, coefficient of kurtosis $(\beta_{\phi_{2}})$ and $\alpha^\prime$ is a scalar $(0\leq\alpha\leq)$ for designing different estimators.

Putting $\alpha^\prime = 1$ and $\alpha^\prime = 0$ in above suggested class of estimators, we get the following estimators.

For $\alpha^\prime = 1$ , the suggested class of estimators reduces to:

$\begin{equation*} t_{RPR(\alpha^\prime = 1)}^\prime = k_{1}^\prime\bar{y}-k_{2}^\prime( p_{1}-p_{1}^\prime) \left[2- exp \left\lbrace \frac{\eta(p_{2}^\prime-P_{2})}{\eta( p_{2}^\prime-P_{2})+2\lambda^\prime} \right\rbrace\right]. \end{equation*}$

For $\alpha^\prime = 0$ , the suggested class of estimators reduces to:

$\begin{equation*} t_{RPR(\alpha^\prime = 0)}^\prime = k_{1}^\prime\bar{y}-k_{2}^\prime( p_{1}-p_{1}^\prime) \left[ exp \left\lbrace \frac{\eta(P_{2}-p_{2}^\prime)}{\eta( P_{2}-p_{2}^\prime)+2\lambda^\prime} \right\rbrace\right]. \end{equation*}$

Expressing (2.32) in terms of errors we have,

$\begin{equation} t_{RPR}^\prime = k_{1}^\prime\bar{Y}(1+e_{0})-k_{2}^\prime P_{1}e_{1}+k_{2} ^\prime P_{1}e_{1}^\prime\bigg[\alpha\left\lbrace 2-(1+\gamma^\prime e_{2}^\prime-\frac{1}{2} \gamma^{2\prime} e^{2\prime}_{2})\right\rbrace+ ( 1-\alpha^\prime)( 1-\gamma^\prime e_{2}^\prime+\frac{3}{2}\gamma^{2\prime} e^{2\prime}_{2})\bigg], \end{equation}$

(2.33)

where $\gamma^\prime = \frac{\eta^\prime P_{2}}{2(\eta^\prime P_{2}+\lambda^\prime)}$ .

To the first degree of approximation,

$\begin{equation*} t_{RPR}^\prime-\bar{Y}\cong k_{1}^\prime \bar{Y}+k_{1}^\prime \bar{Y}e_{0}-k_{2}^\prime P_{1}e_{1}+k_{2}^\prime P_{1}e_{1}^\prime-k_{1}^\prime\gamma^\prime \bar{Y}e_{2}^\prime-\gamma ^\prime\bar{Y} k_{1}^\prime e_{2}^\prime e_{0} \end{equation*}$

$\begin{equation} +\frac{\bar{Y}k_{1}^\prime e^{2\prime}_{2} \gamma^{2\prime} 3}{2}-\alpha^\prime \bar{Y} k_{1}^\prime e^{2\prime}_{2} \gamma^{2\prime}+\gamma k_{2}^\prime P_{1} e_{1}e_{2}^\prime-\gamma k_{2}^\prime P_{1} e_{1}^\prime e_{2}^\prime-\bar{Y}. \end{equation}$

(2.34)

Taking expectation both sides of Eq (2.34) we have:

$\begin{equation} Bias(t_{RPR}^\prime)\cong\bar{Y}(k_{1}^\prime-1)-\gamma\prime\bar{Y}k_{1}^\prime V_{101}^\prime+\bar{Y}k_{1}^\prime V_{002}^\prime\gamma^{2\prime}(\frac{3}{2}-\alpha^\prime). \end{equation}$

(2.35)

Squaring Eq (2.34) and neglecting higher powers, we get

$\begin{equation*} E(t_{RPR}^\prime-\bar{Y}^2)\cong\bar{Y}^2+k_{1}^{\prime2}\bar{Y}^2\left( 1-4\gamma^\prime V_{101}^\prime-2\alpha^\prime V_{002}^\prime\gamma^{2\prime}+4\gamma^2V_{002}^\prime+V_{200}\right) \end{equation*}$

$\begin{equation*} +k_{1}^\prime\bar{Y}^2\left( -2+2\alpha^\prime V_{002}^\prime\gamma^{2\prime}+2\gamma^\prime V_{101}^\prime+3V_{002}^\prime\gamma^{2\prime}\right) \end{equation*}$

$\begin{equation*} +2k_{1}^\prime k_{2}^\prime\bar{Y}\left( P_{1}V_{110}-P_{1}V_{110}^\prime\right)+2k_{2}^2\prime\left( P_{1}^2V_{020}-P_{1}^2V_{020}^\prime\right). \end{equation*}$

$\begin{equation} \text{MSE}(t_{RPR}^\prime)\cong\bar{Y}^2+k_{1}^2\bar{Y}^2 A^\prime+k_{1}\bar{Y}^2B^\prime+2k_{1}k_{2}\bar{Y}C^\prime+k_{2}^2D^\prime, \end{equation}$

(2.36)

$\begin{equation*} A^\prime = 1-4\gamma V_{101}^\prime-2\alpha^\prime V_{002}^\prime\gamma^{2\prime}+4\gamma^2V_{002}^\prime+V_{200}, \end{equation*}$

$\begin{equation*} B^\prime = k_{1} -2+2\alpha^\prime V_{002}^\prime\gamma^{2\prime}+2\gamma^\prime V_{101}^\prime+3V_{002}^\prime\gamma^{2\prime}, \end{equation*}$

$\begin{equation*} C^\prime = P_{1}V_{110}-P_{1}V_{110^\prime}, D^\prime = P_{1}^2V_{020}-P_{1}^2V_{020}^\prime. \end{equation*}$

The optimum values of $k_{1}^\prime$ , $k_{2}^\prime$ are obtained by minimizing Eq (2.36):

$\begin{equation*} k_{1}^\prime = \frac{D^\prime B^\prime}{\left( A^\prime D^\prime-C^{2\prime}\right)}, \end{equation*}$

$\begin{equation*} k_{2}^\prime = \frac{\bar{Y}\left( B^\prime C^\prime\right)}{2\left( A^\prime D^\prime-C^{2\prime}\right)}. \end{equation*}$

Substituting the optimum values of $k_{1}^\prime$ and $k_{2}^\prime$ in Eq $(2.36)$ we get the minimum $\text{MSE}$ of $t_{RPR}^\prime$ as:

$\begin{equation} \text{MSE}(t_{(RPR) min}^\prime)\cong\frac{\bar{Y}^2\left( 1-B^2\prime D^\prime\right)}{4\left(A^\prime D^\prime-C^{2\prime}\right)}. \end{equation}$

(2.37)

3. Efficiency comparisons

3.1. Efficiency comparisons in simple random sampling

In this section we compare theoretically the minimum $\text{MSE}$ of the proposed parent family of estimators $t_{RPR}$ with the $\text{MSE}$ of existing estimators.

Comparison with usual mean per unit estimator:

(i) $\text{MSE}(t_{U})-\text{MSE}(t_{(RPRi)min}) \geq 0 (i = 1, 2, ..., 8)$ , if $\bar{Y}^2V_{200}-[\frac{\bar{Y}^2\left(1-4AD^2+B^2E-4BCD\right)}{4\left(AE-C^2\right)}] \geq 0$ ,

Comparison with Naik ^[1] estimators:

(ii) $\text{MSE}(t_{A})-\text{MSE}(t_{(RPR i) min}) \geq 0$ $(i = 1, 2, ..., 8)$ , if $\bar{Y}^2(V_{200}-2V_{110}+V_{020})-[\frac{\bar{Y}^2\left(1-4AD^2+B^2E-4BCD\right)}{4\left(AE-C^2\right)}] \geq 0$ .

(iii) $\text{MSE}(t_{B})-\text{MSE}(t_{(RPR i) min}) \geq 0$ $(i = 1, 2, ..., 8)$ , if $\bar{Y}^2(V_{200}+2V_{101}+V_{002})-[\frac{\bar{Y}^2\left(1-4AD^2+B^2E-4BCD\right)}{4\left(AE-C^2\right)}] \geq 0$ .

(iv) $\text{MSE}(t_{C})-\text{MSE}(t_{(RPR i) min}) \geq 0$ $(i = 1, 2, ..., 8)$ , if $\bar{Y}^2(V_{200}-V_{110}+\frac{1}{4}V_{020})-[\frac{\bar{Y}^2\left(1-4AD^2+B^2E-4BCD\right)}{4\left(AE-C^2\right)}] \geq 0$ .

(v) $\text{MSE}(t_{D})-\text{MSE}(t_{(RPR i) min}) \geq 0$ $(i = 1, 2, ..., 8)$ , if $\bar{Y}^2(V_{200}+V_{101}+\frac{1}{4}V_{002})-\frac{\bar{Y}^2\left(1-4AD^2+B^2E-4BCD\right)}{4\left(AE-C^2\right)} \geq 0$ .

(vi) $\text{MSE}(t_{MS})-\text{MSE}(t_{(RPR i) min}) \geq 0$ $(i = 1, 2, ..., 8)$ , if $f\bar{Y}^2C_{y}^2(1-R_{y\phi_{1}\phi_{2}}^2)-\frac{\bar{Y}^2\left(1-4AD^2+B^2E-4BCD\right)}{4\left(AE-C^2\right)}\geq 0$ .

We observed that the proposed estimators perform better than the existing estimators if above condition (i)–(vi) are satisfied.

3.2. Efficiency comparisons in two phase sampling

In this section we compare theoretically the minimum $\text{MSE}$ of the proposed parent family of estimators $t_{RPR}$ with the $\text{MSE}$ of existing estimators.

Comparison with usual mean per unit estimator:

(i) $\text{MSE} (t_{U}^\prime) - \text{MSE}(t_{(RPRi)min}^\prime) \geq 0 (i = 1, 2, .., 8)$ , if $\bar{Y}^2(V_{200})-\frac{\bar{Y}^2\left(1-B^{2\prime} D^\prime\right)}{4\left(A^\prime D^\prime-C^{2\prime}\right)} \geq 0.$

Comparison with Naik ^[1] estimator:

(ii) $\text{MSE}(t_{A}^\prime) - \text{MSE}(t_{(RPR i) min}^\prime) \geq 0$ $(i = 1, 2, ..., 8)$ , if $\bar{Y}^2\left(V_{200}+V_{020}-V_{020}^\prime+2V_{110}^\prime-2V_{110}\prime\right) -\frac{\bar{Y}^2\left(1-B^{2\prime} D^\prime\right)}{4\left(A^\prime D^\prime-C^{2\prime}\right)}\geq 0$ .

(iii) $\text{MSE}(t_{B}^\prime) - \text{MSE}(t_{(RPR i) min}^\prime) \geq$ 0 $(i = 1, 2, ..., 8)$ , if $\bar{Y}^2\left(V_{200}+V_{002}^\prime+2V_{101}^\prime\right)-\frac{\bar{Y}^2\left(1-B^{2\prime} D^\prime\right)}{4\left(A^\prime D^\prime-C^{2\prime}\right)}\geq 0$ .

(iv) $\text{MSE}(t_{C}^\prime) - \text{MSE}(t_{(RPR i) min}^\prime) \geq 0$ $(i = 1, 2, ..., 8)$ , if $\bar{Y}^2\left(V_{200}+V_{110}\prime-V_{110}-\frac{1}{4}V_{020}\prime+\frac{1}{4}V_{020}\right)-\frac{\bar{Y}^2\left(1-B^{2\prime} D^\prime\right)}{4\left(A^\prime D^\prime-C^{2\prime}\right)}\geq 0$ .

(v) $\text{MSE}(t_{D}^\prime) - \text{MSE}(t_{(RPR i) min}^\prime) \geq 0$ $(i = 1, 2, ..., 8)$ , if $\bar{Y}^2\left(V_{200}+\frac{1}{4}V_{002}^\prime+V_{101}^\prime\right)-\frac{\bar{Y}^2\left(1-B^{2\prime} D^\prime\right)}{4\left(A^\prime D^\prime-C^{2\prime}\right)}\geq 0$ .

(vi) $\text{MSE}(t_{MS}^\prime) - \text{MSE}(t_{(RPR i) min}^\prime) \geq 0$ $(i = 1, 2, ..., 8)$ , if $f\bar{Y}^2C_{y}^2\left\lbrace -f(-1+\rho_{y\phi_{1}}^2)+ \lambda(\rho_{y\phi_{1}}^2-\rho_{y\phi_{2}}^2)\right\rbrace -\frac{\bar{Y}^2\left(1-B^{2\prime} D^\prime\right)}{4\left(A^\prime D^\prime-C^{2\prime}\right)}\geq 0$ .

We observed that the proposed estimators perform better than the existing estimators if above condition (i)–(vi) are satisfied.

4. Numerical study

4.1. Numerical study in simple random sampling

Population 1. ^[19]

Let $Y$ be the study variable of the cultivated area of wheat in 1964.

$P_{1}$ be the proportion of cultivated area of wheat greater than 100 acre in 1963.

$P_{2}$ be the proportion of cultivated area of wheat greater than 500 in 1961.

$N = 34$ , $n = 15$ , $\bar{Y} = 199.4412$ , $P_{1} = 0.73529$ , $P_{2} = 0.647059$ , $S_{y} = 150.215$ , $S_{\phi_{1}} = 0.4478111$ , $S_{\phi_{2}} = 0.4850713$ , $\beta_{2\phi_{2}} = -1.688$ , $C_{\phi_{1}} = 0.6090231$ , $\rho_{\phi_{1}\phi_{2}} = 0.6729$ , $C_{\phi_{2}} = 0.7496556$ , $C_{y} = 0.7531$ ,

$\rho_{y}\phi_{2} = 0.6281$ , $\rho_{y}\phi_{1} = 0.559$ .

Population 2. ^[20]

Let $Y$ be the study variable of the number of fishes caught in 1995.

$P_{1}$ be the proportion of fishes caught which is greater than 1000 in 1993.

$P_{2}$ be the proportion of fishes caught which is greater than 2000 in 1994.

$N = 69$ , $n = 14$ , $\bar{Y} = 4514.89$ , $P_{1} = 0.7391304$ , $P_{2} = 0.5507246$ , $S_{y} = 6099.14$ , $S_{\phi_{1}} = 0.4423259$ , $S_{\phi_{2}} = 0.5010645$ , $\beta_{2\phi_{2}} = -2.015$ , $C_{\phi_{1}} = 0.5984409$ , $\rho_{\phi_{1}\phi_{2}} = 0.6577519$ , $C_{\phi_{2}} = 0.9098277$ , $C_{y} = 1.350$ , $\rho_{y\phi_{2}} = 0.538047$ , $\rho_{y\phi_{1}} = 0.3966081$ .

Population 3. ^[21]

Let study variable $Y$ be the tobacco area production in hectares during the year $2009$ .

$P_{1}$ be the proportion of farms with tobacco cultivation area greater than 500 hectares during the year $2007$ .

$P_{2}$ be proportion of farms with tobacco cultivation area greater than 800 hectares during the year $2008$ for 47 districts of Pakistan.

$N = 47$ , $n = 10$ , $\bar{Y} = 1004.447$ , $P_{1} = 0.4255319$ , $P_{2} = 0.3829787$ , $s_{y} = 2351.656$ , $s_{\phi_{1}} = 0.499$ , $s_{\phi_{2}} = 0.4850713$ , $\beta_{2\phi_{2}} = -1.8324$ , $C_{\phi_{1}} = 1.174456$ , $\rho_{\phi_{1}\phi_{2}} = 0.9153857$ , $C_{\phi_{2}} = 1.283018$ , $C_{y} = 2.341245$ , $\rho_{y\phi_{2}} = 0.4661508$ , $\rho_{y\phi_{1}} = 0.4395989$ .

Population 4. ^[21]

Let study variable $Y$ be the cotton production in hectares during the year 2009.

$P_{1}$ be the proportion of farms with cotton cultivation area greater than 37 hectares during the year 2007.

$P_{2}$ be proportion of farms with cotton cultivation area greater than 35 hectares during the year 200 for 52 districts of Pakistan.

$N = 52$ , $n = 11$ , $\bar{Y} = 50.03846$ , $P_{1} = 0.3846154$ , $P_{2} = 0.4423077$ , $S_{y} = 71.13086$ , $S_{\phi_{1}} = 0.4912508$ , $S_{\phi_{2}} = 0.501506$ , $\beta_{2\phi_{2}} = -1.62014$ , $C_{\phi_{1}} = 1.277252$ , $\rho_{\phi_{1}\phi_{2}} = 0.8877181$ , $C_{\phi_{2}} = 1.13384$ , $C_{y} = 1.421524$ , $\rho_{y\phi_{2}} = 0.6935718$ , $\rho_{y\phi_{1}} = 0.7369579$ .

We use the following expression to obtain the Percentage Relative Efficiency $\text{PRE}$ :

$\begin{equation} PRE = \frac {\text{MSE}(t_{0})}{\text{MSE}(t_{i}min)}*100, \end{equation}$

(4.1)

where i = U, A, B, C, D, MS, RPR1, RPR2, RPR3, RPR4, RPR5, RPR6, RPR7 and RPR8.

In , it is clearly shown that our suggested class of estimator $t_{RPR_{i}}$ perform better than all the existing estimators $t_{A}$ , $t_{B}$ , $t_{C}$ , $t_{D}$ and $t_{MS}$ . A significant increase is observed in the percentage relative efficiency of estimators of $t_{RPR6}$ , $t_{RPR7}$ and $t_{RPR8}$ .

Table 2. Set of estimators generated from estimator

$t_{RPR(\alpha^\prime = 1)}^\prime$ .

$\text{Subset of proposed estimator}$	$\alpha^\prime$	$\eta^\prime$	$\lambda^\prime$
$t_{RPR1}^\prime=k_{1}^\prime\bar{y}-k_{2}^\prime(p_{1}-p_{1}^\prime) \left[exp \left\lbrace \frac{C_{\phi_{2}}(P_{2}-p_{2}^\prime)}{C_{\phi_{2}}(P_{2}-p_{2}^\prime)+2\beta_{2}\phi_{2}} \right\rbrace\right]$	0	$C_{\phi_{2}}$	$\beta_{2\phi_{2}}$
$t_{RPR2}^\prime=k_{1}^\prime\bar{y}-k_{2}^\prime(p_{1}-p_{1}^\prime) \left[exp\left\lbrace \frac{P_{2}(P_{2}-p_{2}^\prime)}{P_{2}(P_{2}-p_{2}^\prime)+2} \right\rbrace\right]$	0	$P_{2}$	$1$
$t_{RPR3}^\prime=k_{1}^\prime\bar{y}-k_{2}^\prime(p_{1}-p_{1}^\prime) \left[exp \left\lbrace \frac{(P_{2}-p_{2}^\prime)}{(P_{2}-p_{2}^\prime)+2C_{\phi_{2}}} \right\rbrace\right]$	0	1	$C_{\phi_{2}}$
$t_{RPR4}^\prime=k_{1}\bar{y}-k_{2}(p_{1}-p_{1}^\prime)\left[exp \left\lbrace \frac{(P_{2}-p_{2}^\prime)}{(P_{2}-p_{2}^\prime)+2} \right\rbrace\right]$	0	1	1
$t_{RPR5}^\prime=k_{1}^\prime\bar{y}-k_{2}^\prime(p_{1}-p_{1}^\prime)\left[2-exp\left\lbrace \frac{C_{\phi_{2}}(p_{2}^\prime-P_{2})}{C_{\phi_{2}}(p_{2}-P_{2}^\prime)+2\beta_{2}\phi_{2}} \right\rbrace\right]$	1	$C_{\phi_{2}}$	$\beta_{2\phi_{2}}$
$t_{RPR6}^\prime=k_{1}^\prime\bar{y}-k_{2}^\prime(p_{1}-p_{1}^\prime)\left[2- exp\left\lbrace \frac{P_{2}(p_{2}^\prime-P_{2})}{P_{2}(p_{2}^\prime-P_{2})+2} \right\rbrace\right]$	1	$P_{2}$	$1$
$t_{RPR7}^\prime=k_{1}^\prime\bar{y}-k_{2}^\prime(p_{1}-p_{1}^\prime)\left[2-exp\left\lbrace\frac{(p_{2}^\prime-P_{2})}{(p_{2}^\prime-P_{2})+2C_{\phi_{2}}}\right\rbrace\right]$	1	1	$C_{\phi_{2}}$
$t_{RPR8}^\prime=k_{1}^\prime\bar{y}-k_{2}^\prime(p_{1}-p_{1}^\prime)\left[2-exp\left\lbrace \frac{(p_{2}^\prime-P_{2})}{(p_{2}^\prime-P_{2})+2}\right\rbrace\right]$	1	1	1

| Show Table

DownLoad: CSV

4.2. Empirical study in two phase sampling

Population 1. ^[19]

Let $Y$ be the study variable cultivated area of wheat in 1964.

$P_{1}$ be the proportion of cultivated area of wheat greater than 100 acres in 1963.

$P_{2}$ be the proportion of cultivated area of wheat greater than 500 in 1961.

$N = 34$ , $n^\prime = 15$ , $n = 3$ , $\bar{Y} = 199.4412$ , $P_{1} = 0.73529$ , $P_{2} = 0.647059$ , $S_{y} = 150.215$ , $S_{\phi_{1}} = 0.4478111$ , $S_{\phi_{2}} = 0.4850713$ , $\beta_{2\phi_{2}} = -1.688$ , $C_{\phi_{1}} = 0.6090231$ , $\rho_{\phi_{1}\phi_{2}} = 0.6729$ , $C_{\phi_{2}} = 0.7496556$ , $C_{y} = 0.7531$ , $\rho_{y\phi_{2}} = 0.6281$ , $\rho_{y\phi_{1}} = 0.559$ .

Population 2. ^[20]

Let $Y$ be the study variable, number of fishes caught in 1995.

$P_{1}$ be the proportion of fishes caught greater than 1000 in 1993.

$P_{2}$ be the proportion of fishes caught greater than 2000 in 1994.

$N = 69$ , $n^\prime = 20$ , $n = 7$ , $\bar{Y} = 4514.89$ , $P_{1} = 0.7391304$ , $P_{2} = 0.5507246$ , $s_{y} = 6099.14$ , $s_{\phi_{1}} = 0.4423259$ , $s_{\phi_{2}} = 0.5010645$ , $\beta_{2\phi_{2}} = -2.015$ , $C_{\phi_{1}} = 0.5984409$ , $\rho_{\phi_{1}\phi_{2}} = 0.6577519$ , $C_{\phi_{2}} = 0.9098277$ , $C_{y} = 1.350$ , $\rho_{y\phi_{2}} = 0.538047$ , $\rho_{y\phi_{1}} = 0.3966081$ .

Population 3. ^[21]

Let $Y$ be the study variable, tobacco area production in hectares during the year $2009$ .

$P_{1}$ be the proportion of farms with tobacco cultivation area greater than 500 hectares during the year $2007$ .

$P_{2}$ be proportion of farms with tobacco cultivation area greater than 800 hectares during the year $2008$ for 47 districts of Pakistan.

$N = 47$ , $n^\prime = 15$ , $n = 7$ , $\bar{Y} = 1004.447$ , $P_{1} = 0.4255319$ , $P_{2} = 0.3829787$ , $S_{y} = 2351.656$ , $S_{\phi_{1}} = 0.49$ , $S_{\phi_{2}} = 0.4850713$ , $\beta_{2\phi_{2}} = -1.8324$ , $C_{\phi_{1}} = 1.174456$ , $\rho_{\phi_{1}\phi_{2}} = 0.9153857$ , $C_{\phi_{2}} = 1.283018$ , $C_{y} = 2.341245$ , $\rho_{y\phi_{2}} = 0.4661508$ , $\rho_{y\phi_{1}} = 0.4395989$ .

Population 4. ^[21]

Let $Y$ be the study variable, cotton production in hectares during the year $2009$ .

$P_{1}$ be the proportion of farms with cotton cultivation area greater than 37 hectares during the year $2007$ .

$P_{2}$ be proportion of farms with cotton cultivation area greater than 35 hectares during the year $2008$ for 52 districts of Pakistan.

$N = 52$ , $n^\prime = 11$ , $n = 3$ , $\bar{Y} = 50.03846$ , $P_{1} = 0.3846154$ , $P_{2} = 0.4423077$ , $S_{y} = 71.13086$ , $S_{\phi_{1}} = 0.4912508$ , $S_{\phi_{2}} = 0.501506$ , $\beta_{2\phi_{2}} = -1.62014$ , $C_{\phi_{1}} = 1.277252$ , $\rho_{\phi_{1}\phi_{2}} = 0.8877181$ , $C_{\phi_{2}} = 1.13384$ , $C_{y} = 1.421524$ , $\rho_{y\phi_{2}} = 0.6935718$ , $\rho_{y\phi_{1}} = 0.7369579$ .

We use the following expression to obtain the Percentage Relative Efficiency(PRE):

$\begin{equation} PRE = \frac {\text{MSE}(t_{0})}{\text{MSE}(t_{i min}^\prime)}*100, \end{equation}$

(4.2)

where i = $U^\prime$ , $A^\prime, B^\prime, C^\prime, D^\prime, MS^\prime, RPR1^\prime, RPR2^\prime, RPR3^\prime, RPR4^\prime, RPR5^\prime, RPR6^\prime, RPR7^\prime and RPR8^\prime.$

The results for data set 1–4 are given in Table 4.

In , it is clearly shown that our suggested class of estimator $t_{RPR}^\prime$ perform better than all the existing estimators of $t_{A}^\prime$ , $t_{B}^\prime$ , $t_{C}^\prime$ and $t_{D}^\prime$ and $t_{MS}^\prime$ . A significant increase is observed in the percentage relative efficiency of estimators of $t_{RPR6}^\prime$ , $t_{RPR7}^\prime$ and $t_{RPR8}^\prime$ .

There are many situations where we only interest in knowing everything about the study variable, which is too difficult. For this we can use two auxiliary variables in the form of proportion to find out the study variable. This manuscript provides us the basic tools to the problems related to proportion estimation and two-phase sampling. Here we can see that in abstract of the manuscript we just talk about the minimum MSE of proposed and existing estimators, reason behind is that we can easily compare the minimum MSE with other properties of good estimators like MLE ect., we can also see that the comparison is made in the form of percentage relative efficiency.

5. Application

Statisticians are constantly trying to develop efficient estimators and estimation methodologies to increase the efficiency of estimates. The progress is going on for estimators of population mean. In the present paper our task is to develop a new estimator for estimating the finite population mean under two different sampling schemes, which are simple random sampling and two-phase sampling. The new estimators will be proposed under the following situations:

1). The initial sample is collected through simple random sampling.

2). And then by two-phase sampling using simple random sampling.

In this article, we consider the problem of estimating the finite population mean using the auxiliary proportion under simple random sampling and two-phase sampling scheme. In general, during surveys, it is observed that information in most cases is not obtained on the first attempt even after some call-backs, in such types of issue we use simple random sampling. And when the required results are not obtained, we use two-phase sampling. These approaches are used to obtain the information as much as possible. In sample surveys, it is well known that while estimating the population parameters, i.e., Finite population (mean, median, quartiles, coefficient of variation and distribution function) the information of the auxiliary variable (Proportion) is usually used to improve the efficiency of the estimators. The main aim of studies is to find out more efficient estimators than classical and recent proposed estimators using the auxiliary information (in the form of proportion) for estimating finite population mean under simple random sampling and two-phase sampling scheme.

There are situations where our work is deemed necessary and can be used in daily life.

1). For a nutritionist, it is interesting to know the proportion of population that consumes 25% or more of the calorie intake from saturated fat.

2). Similarly, a soil scientist may be interested in estimating the distribution of clay percent in the soil.

3). In addition, policy-makers may be interested in knowing the proportion of people living in a developing country below the poverty line.

6. Conclusions

In this paper, we have proposed a generalized class of exponential ratio type estimators for estimating population mean using the auxiliary information in the form of proportions under simple and two phase sampling. We used $\text{SRS}$ to estimate the population mean using the proportions of available auxiliary information, and when the auxiliary information is unknown, we used two phase sampling for estimation resolution. From the numerical results available in and we can see that two phase sampling gave more efficient results than simple random sampling. Thus the use of auxiliary information in estimation processes increases the efficiency of the estimator, that's we have used two auxiliary variables as attributes. In the numerical study we showed that the proposed estimator is more efficient that $t_{U}$ , $t_{A}$ , $t_{B}$ , $t_{C}$ , $t_{D}$ , $t_{MS}$ and any other suggested family of estimators both in simple and two phase sampling schemes.

Table 3. Percentage relative efficiency (PRE) with respect to usual mean estimator

$t_{U}$ .

Estimator	Data set1	Data set 2	Data set 3	Data set 4
$t_{U}$	100	100	100	100
$t_{A}$	133.37	118.36	123.36	207.04
$t_{B}$	30.84	45.95	55.04	36.46
$t_{C}$	140.5	114.40	118.75	185.29
$t_{D}$	55.39	67.77	75.01	58.40
$t_{MS}$	139.06	110.94	105.64	146.93
$t_{RPR1}$	125.98	134.42	165.4	225.72
$t_{RPR2}$	106.66	134.57	167.49	235.43
$t_{RPR3}$	111.08	137.09	167.89	235.65
$t_{RPR4}$	109.10	137.39	167.82	233.82
$t_{RPR5}$	125.75	120.83	166.47	223.72
$t_{RPR6}$	161.80	137.02	167.63	235.16
$t_{RPR7}$	168.29	137.47	168.16	235.96
$t_{RPR8}$	165.09	134.42	168.01	235.93

| Show Table

DownLoad: CSV

Table 4. Percentage relative efficiency (PRE) with respect to usual mean estimator

$t_{U}^\prime$ .

Estimator	Data set1	Data set 2	Data set 3	Data set 4
$t_{U}^\prime$	100	100	100	100
$t_{A}^\prime$	128.13	112.64	113.46	166.39
$t_{B}^\prime$	78.44	75.41	76.75	71.54
$t_{C}^\prime$	133.90	110.08	110.95	155.10
$t_{D}^\prime$	90.33	88.37	89.01	86.01
$t_{MS}^\prime$	134.80	111.05	133.6	209.34
$t_{RPR1}^\prime$	149.36	131.69	178.24	225.68
$t_{RPR2}^\prime$	158.43	138.50	181.06	239.47
$t_{RPR3}^\prime$	160.04	139.77	181.55	242.17
$t_{RPR4}^\prime$	159.63	139.58	181.78	242.08
$t_{RPR5}^\prime$	149.59	131.98	178.95	227.06
$t_{RPR6}^\prime$	158.56	138.59	181.13	239.69
$t_{RPR7}^\prime$	160.36	139.77	181.13	242.80
$t_{RPR8}^\prime$	159.40	139.99	182.10	242.70

| Show Table

DownLoad: CSV

Some possible extensions of the current work are as follows:

Develop improved finite population mean estimators,

1). using supplementary information more than one auxiliary variable.

2). under stratified two-phase sampling.

3). in the presence of measurement errors.

4). under non-response with two-phase sampling.

Acknowledgments

The authors are thankful to the learned referee for his useful comments and suggestions.

Conflict of interest

The authors declare no conflict of interest.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

References

[1]	Bhugra D (2005) The global prevalence of schizophrenia. PLoS Med 2: e151. doi: 10.1371/journal.pmed.0020151
[2]	Friston KJ (1998) The disconnection hypothesis. Schizophr Res 30: 115-125. doi: 10.1016/S0920-9964(97)00140-0
[3]	Liu Y, Liang M, Zhou Y, et al. (2008) Disrupted small-world networks in schizophrenia. Brain 131: 945-961. doi: 10.1093/brain/awn018
[4]	Lynall ME, Bassett DS, Kerwin R, et al. (2010) Functional connectivity and brain networks in schizophrenia. J Neurosci 30: 9477-9487. doi: 10.1523/JNEUROSCI.0333-10.2010
[5]	Griffa A, Baumann PS, Ferrari C, et al. (2015) Characterizing the connectome in schizophrenia with diffusion spectrum imaging. Hum Brain Mapp 36: 354-366. doi: 10.1002/hbm.22633
[6]	Biswal BB (2012) Resting state fMRI: a personal history. Neuroimage 62: 938-944. doi: 10.1016/j.neuroimage.2012.01.090
[7]	Dong D, Wang Y, Chang X, et al. (2018) Dysfunction of large-scale brain networks in schizophrenia: a meta-analysis of resting-state functional connectivity. Schizophr Bull 44: 168-181. doi: 10.1093/schbul/sbx034
[8]	Smith SM, Beckmann CF, Andersson J, et al. (2013) Resting-state fMRI in the human connectome project. Neuroimage 80: 144-168. doi: 10.1016/j.neuroimage.2013.05.039
[9]	Moussa MN, Steen MR, Laurienti PJ, et al. (2012) Consistency of Network Modules in Resting-State fMRI Connectome Data. PloS One 7: e44428. doi: 10.1371/journal.pone.0044428
[10]	Smith SM, Fox PT, Miller KL, et al. (2009) Correspondence of the brain's functional architecture during activation and rest. Proce Natl Acad Sci 106: 13040-13045. doi: 10.1073/pnas.0905267106
[11]	Damoiseaux JS, Rombouts SARB, Barkhof F, et al. (2006) Consistent resting-state networks across healthy subjects. Proce Natl Acad Sci 103: 13848-13853. doi: 10.1073/pnas.0601417103
[12]	Argyelan M, Ikuta T, DeRosse P, et al. (2014) Resting-state fMRI connectivity impairment in schizophrenia and bipolar disorder. Schizophr Bull 40: 100-110. doi: 10.1093/schbul/sbt092
[13]	Arribas JI, Calhoun VD, Adali T (2010) Automatic Bayesian classification of healthy controls, bipolar disorder, and schizophrenia using intrinsic connectivity maps from FMRI data. IEEE Transa Biomed Eng 57: 2850-2860. doi: 10.1109/TBME.2010.2080679
[14]	Matsubara T, Tashiro T, Uehara K (2019) Deep neural generative model of functional MRI images for psychiatric disorder diagnosis. IEEE Transa Biomed Eng 66: 2768-2779. doi: 10.1109/TBME.2019.2895663
[15]	Xiang Y, Wang J, Tan G, et al. (2020) Schizophrenia identification using multi-view graph measures of functional brain networks. Front Bioeng Biotechnol 7: 479. doi: 10.3389/fbioe.2019.00479
[16]	Anderson A, Cohen MS (2013) Decreased small-world functional network connectivity and clustering across resting state networks in schizophrenia: an fMRI classification tutorial. Front Hum Neurosci 7: 520.
[17]	Arbabshirani MR, Kiehl KA, Pearlson DG (2013) Classification of schizophrenia patients based on resting-state functional network connectivity. Front Neurosci 7: 133. doi: 10.3389/fnins.2013.00133
[18]	Cheng H, Newman S, Goñi J, et al. (2015) Classification of schizophrenia patients based on restingstate functional network connectivity. Schizophr Res 168: 345-352. doi: 10.1016/j.schres.2015.08.011
[19]	Moghimi P, Lim KO, Netoff TI (2018) Data Driven Classification Using fMRI Network Measures: Application to Schizophrenia. Front Neuroinform 12: 71. doi: 10.3389/fninf.2018.00071
[20]	Sundermann B, Herr D, Schwindt W, et al. (2014) Multivariate classification of blood oxygen level–dependent fMRI data with diagnostic intention: a clinical perspective. Am J Neuroradiol 35: 848-855. doi: 10.3174/ajnr.A3713
[21]	Cai XL, Xie DJ, Madsen KH, et al. (2020) Generalizability of machine learning for classification of schizophrenia based on resting-state functional MRI data. Hum Brain Mapp 41: 172-184. doi: 10.1002/hbm.24797
[22]	Du W, Calhoun VD, Li H, et al. (2012) High classification accuracy for schizophrenia with rest and task fMRI data. Front Hum Neurosci 6: 145.
[23]	Pereira F, Mitchell T, Botvinick M (2009) Machine learning classifiers and fMRI: a tutorial overview. Neuroimage 45: S199-S209. doi: 10.1016/j.neuroimage.2008.11.007
[24]	Krstajic D, Buturovic LJ, Leahy DE, et al. (2014) Cross-validation pitfalls when selecting and assessing regression and classification models. J Cheminform 6: 1-15. doi: 10.1186/1758-2946-6-10
[25]	Pettersson-Yeo W, Allen P, Benetti S, et al. (2011) Dysconnectivity in schizophrenia: where are we now? Neurosci Biobehav Rev 35: 1110-1124. doi: 10.1016/j.neubiorev.2010.11.004
[26]	Fekete T, Wilf M, Rubin D, et al. (2013) Combining classification with fMRI-derived complex network measures for potential neurodiagnostics. PloS One 8: e62867. doi: 10.1371/journal.pone.0062867
[27]	Gallos IK, Galaris E, Siettos CIConstruction of embedded fMRI resting-state functional connectivity networks using manifold learning. Cogn Neurodyn (2020) .
[28]	Lombardi A, Guaragnella C, Amoroso N, et al. (2019) Modelling cognitive loads in schizophrenia by means of new functional dynamic indexes. NeuroImage 195: 150-164. doi: 10.1016/j.neuroimage.2019.03.055
[29]	Hyv¨arinen A, Oja E (2000) Independent component analysis: algorithms and applications. Neural Netw 13: 411-430. doi: 10.1016/S0893-6080(00)00026-5
[30]	Haak KV, Marquand AF, Beckmann CF (2018) Connectopic mapping with resting-state fMRI. Neuroimage 170: 83-94. doi: 10.1016/j.neuroimage.2017.06.075
[31]	Desikan RS, Ségonne F, Fischl B, et al. (2006) An automated labeling system for subdividing the human cerebral cortex on MRI scans into gyral based regions of interest. Neuroimage 31: 968-980. doi: 10.1016/j.neuroimage.2006.01.021
[32]	Kabbara A, Falou WEL, Khalil M, et al. (2017) The dynamic functional core network of the human brain at rest. Sci Rep 7: 1-16. doi: 10.1038/s41598-017-03420-6
[33]	Tenenbaum JB, De Silva V, Langford JC (2000) A global geometric framework for nonlinear dimensionality reduction. Science 290: 2319-2323. doi: 10.1126/science.290.5500.2319
[34]	Liu J, Li M, Pan Y, et al. (2017) Complex brain network analysis and its applications to brain disorders: a survey. Complexity 2017: 1-27.
[35]	Jenkinson M, Bannister P, Brady M, et al. (2002) Improved optimization for the robust and accurate linear registration and motion correction of brain images. Neuroimage 17: 825-841. doi: 10.1006/nimg.2002.1132
[36]	Smith SM (2002) Fast robust automated brain extraction. Hum Brain Mapp 17: 143-155. doi: 10.1002/hbm.10062
[37]	Pruim RHR, Mennes M, van Rooij D, et al. (2015) ICA-AROMA: A robust ICA-based strategy for removing motion artifacts from fMRI data. Neuroimage 112: 267-277. doi: 10.1016/j.neuroimage.2015.02.064
[38]	Kalcher K, Boubela RN, Huf W, et al. (2014) The spectral diversity of resting-state fluctuations in the human brain. PloS One 9: e93375. doi: 10.1371/journal.pone.0093375
[39]	Fischl B (2012) FreeSurfer. Neuroimage 62: 774-781. doi: 10.1016/j.neuroimage.2012.01.021
[40]	Hyde JS, Jesmanowicz A (2012) Cross-correlation: an fMRI signal-processing strategy. Neuroimage 62: 848-851. doi: 10.1016/j.neuroimage.2011.10.064
[41]	van den Heuvel MP, de Lange SC, Zalesky A, et al. (2017) Proportional thresholding in restingstate fMRI functional connectivity networks and consequences for patient-control connectome studies: Issues and recommendations. Neuroimage 152: 437-449. doi: 10.1016/j.neuroimage.2017.02.005
[42]	Dimitriadis SI, Antonakakis M, Simos P, et al. (2017) Data-driven topological filtering based on orthogonal minimal spanning trees: application to multigroup magnetoencephalography restingstate connectivity. Brain Connect 7: 661-670. doi: 10.1089/brain.2017.0512
[43]	Dimitriadis SI, Salis C, Tarnanas I, et al. (2017) Topological filtering of dynamic functional brain networks unfolds informative chronnectomics: a novel data-driven thresholding scheme based on orthogonal minimal spanning trees (OMSTs). Front Neuroinform 11: 28. doi: 10.3389/fninf.2017.00028
[44]	Dijkstra EW (1959) A note on two problems in connexion with graphs. Front Neuroinform 1: 269-271.
[45]	Kruskal JB (1964) Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika 29: 1-27. doi: 10.1007/BF02289565
[46]	Saul LK, Weinberger KQ, Ham JH, et al. (2006) Spectral methods for dimensionality reduction. Semisupervised Learning Cambridge, MA, USA: The MIT Press, 292-308.
[47]	Oksanen J, Kindt R, Legendre P, et al. (2007) The vegan package. Community Ecol Package 10: 631-637.
[48]	Team, R Core R: A language and environment for statistical computing (2013) .
[49]	Rubinov M, Sporns O (2010) Complex network measures of brain connectivity: uses and interpretations. Neuroimage 52: 1059-1069. doi: 10.1016/j.neuroimage.2009.10.003
[50]	Blondel VD, Guillaume JL, Lambiotte R, et al. (2008) Fast unfolding of communities in large networks. J Stat Mech Theory Exp 2008: P10008. doi: 10.1088/1742-5468/2008/10/P10008
[51]	Barrat A, Barthélemy M, Pastor-Satorras R, et al. (2004) The architecture of complex weighted networks. Proc Natl Acad Sci USA 101: 3747-3752. doi: 10.1073/pnas.0400087101
[52]	Power JD, Schlaggar BL, Lessov-Schlaggar CN, et al. (2013) Evidence for hubs in human functional brain networks. Neuron 79: 798-813. doi: 10.1016/j.neuron.2013.07.035
[53]	Csardi G, Nepusz T (2006) The igraph software package for complex network research. InterJournal Complex Syst 1695: 1-9.
[54]	Chan MK, Krebs MO, Cox D, et al. (2015) Development of a blood-based molecular biomarker test for identification of schizophrenia before disease onset. Transl Psychiatry 5: e601. doi: 10.1038/tp.2015.91
[55]	Usai MG, Goddard ME, Hayes BJ (2009) LASSO with cross-validation for genomic selection. Genet Res 91: 427-436. doi: 10.1017/S0016672309990334
[56]	Friedman J, Hastie T, Tibshirani R (2010) Regularization paths for generalized linear models via coordinate descent. J Stat Softw 33: 1-22. doi: 10.18637/jss.v033.i01
[57]	Hastie T, Tibshirani R, Friedman J (2009) The elements of statistical learning: data mining, inference, and prediction New York City, USA: Springer.
[58]	Breiman L (2001) Random forests. Mach Learn 45: 5-32. doi: 10.1023/A:1010933404324
[59]	Loh WY (2011) Classification and regression trees. WIRES Data Min Knowl 1: 14-23. doi: 10.1002/widm.8
[60]	Menze BH, Kelm BM, Masuch R, et al. (2009) A comparison of random forest and its Gini importance with standard chemometric methods for the feature selection and classification of spectral data. BMC Bioinformatics 10: 213. doi: 10.1186/1471-2105-10-213
[61]	Zhu X, Du X, Kerich M, et al. (2018) Random forest based classification of alcohol dependence patients and healthy controls using resting state MRI. Neurosci Lett 676: 27-33. doi: 10.1016/j.neulet.2018.04.007
[62]	Kesler SR, Rao A, Blayney DW, et al. (2017) Predicting long-term cognitive outcome following breast cancer with pre-treatment resting state fMRI and random forest machine learning. Front Hum Neurosci 11: 555. doi: 10.3389/fnhum.2017.00555
[63]	Ceriani L, Verme P (2012) The origins of the Gini index: extracts from Variabilità e Mutabilità (1912) by Corrado Gini. J Econ Inequal 10: 421-443. doi: 10.1007/s10888-011-9188-x
[64]	Strobl C, Boulesteix AL, Augustin T (2007) Unbiased split selection for classification trees based on the Gini index. Comput Stat Data Anal 52: 483-501. doi: 10.1016/j.csda.2006.12.030
[65]	Louppe G, Wehenkel L, Sutera A, et al. (2013) Understanding variable importances in forests of randomized trees. Adv Neural Inf Process Syst 26: 431-439.
[66]	Behnamian A, Millard K, Banks SN, et al. (2017) A systematic approach for variable selection with random forests: achieving stable variable importance values. IEEE Geosci Remote S 14: 1988-1992. doi: 10.1109/LGRS.2017.2745049
[67]	RColorBrewer S, Liaw A, Wiener M, et al. Package ‘randomForest’ (2018) .
[68]	Kuhn M (2008) Building predictive models in R using the caret package. J Stat Softw 28: 1-26. doi: 10.18637/jss.v028.i05
[69]	Nieuwenhuis M, van Haren NE, Pol HEH, et al. (2012) Classification of schizophrenia patients and healthy controls from structural MRI scans in two large independent samples. Neuroimage 61: 606-612. doi: 10.1016/j.neuroimage.2012.03.079
[70]	Fan Y, Liu Y, Wu H, et al. (2011) Discriminant analysis of functional connectivity patterns on Grassmann manifold. Neuroimage 56: 2058-2067. doi: 10.1016/j.neuroimage.2011.03.051
[71]	Salman MS, Du Y, Lin D, et al. (2019) Group ICA for identifying biomarkers in schizophrenia: ‘Adaptive’ networks via spatially constrained ICA show more sensitivity to group differences than spatio-temporal regression. NeuroImage Clin 22: 101747. doi: 10.1016/j.nicl.2019.101747
[72]	Himberg J, Hyvärinen A, Esposito F (2004) Validating the independent components of neuroimaging time series via clustering and visualization. Neuroimage 22: 1214-1222. doi: 10.1016/j.neuroimage.2004.03.027
[73]	Cole DM, Smith SM, Beckmann CF (2010) Advances and pitfalls in the analysis and interpretation of resting-state FMRI data. Front Syst Neurosci 4: 8.
[74]	Andreasen NC (1997) The role of the thalamus in schizophrenia. Can J Psychiatry 42: 27-33. doi: 10.1177/070674379704200104
[75]	Byne W, Hazlett EA, Buchsbaum MS, et al. (2009) The thalamus and schizophrenia: current status of research. Acta Neuropathol 117: 347-368. doi: 10.1007/s00401-008-0404-0
[76]	Pergola G, Selvaggi P, Trizio S, et al. (2015) The role of the thalamus in schizophrenia from a neuroimaging perspective. Neurosci Biobehav Rev 54: 57-75. doi: 10.1016/j.neubiorev.2015.01.013
[77]	Bogousslavsky J, Miklossy J, Deruaz JP, et al. (1987) Lingual and fusiform gyri in visual processing: a clinico-pathologic study of superior altitudinal hemianopia. J Neurol Neurosurg Psychiatry 50: 607-614. doi: 10.1136/jnnp.50.5.607
[78]	Mechelli A, Humphreys GW, Mayall K, et al. (2000) Differential effects of word length and visual contrast in the fusiform and lingual gyri during reading. Proc Biol Sci 267: 1909-1913. doi: 10.1098/rspb.2000.1229
[79]	Yu T, Li Y, Fan F, et al. (2018) Decreased gray matter volume of cuneus and lingual gyrus in schizophrenia patients with tardive dyskinesia is associated with abnormal involuntary movement. Sci Rep 8: 1-7. doi: 10.1038/s41598-017-17765-5
[80]	Kogata T, lidaka T (2018) A review of impaired visual processing and the daily visual world in patients with schizophrenia. Nagoya J Med Sci 80: 317-328.
[81]	Yamamoto M, Bagarinao E, Kushima I, et al. (2020) Support vector machine-based classification of schizophrenia patients and healthy controls using structural magnetic resonance imaging from two independent sites. PloS One 15: e0239615. doi: 10.1371/journal.pone.0239615
[82]	Tohid H, Faizan M, Faizan U (2015) Alterations of the occipital lobe in schizophrenia. Neurosciences 20: 213-224. doi: 10.17712/nsj.2015.3.20140757

This article has been cited by:

1.	Muhammad Ahmed Shehzad, Anam Nisar, Aamna Khan, Walid Emam, Yusra Tashkandy, Haris Khurram, Isra Al-Shbeil, Modified median quartile double ranked set sampling for estimation of population mean, 2024, 10, 24058440, e34627, 10.1016/j.heliyon.2024.e34627
2.	Muhammad Nadeem Intizar, Muhammad Ahmed Shehzad, Haris Khurram, Soofia Iftikhar, Aamna Khan, Abdul Rauf Kashif, Integrating endogeneity in survey sampling using instrumental-variable calibration estimator, 2024, 10, 24058440, e33969, 10.1016/j.heliyon.2024.e33969
3.	Anoop Kumar, Walid Emam, Yusra Tashkandy, Memory type general class of estimators for population variance under simple random sampling, 2024, 10, 24058440, e36090, 10.1016/j.heliyon.2024.e36090
4.	Jing Wang, Sohaib Ahmad, Muhammad Arslan, Showkat Ahmad Lone, A.H. Abd Ellah, Maha A. Aldahlan, Mohammed Elgarhy, Estimation of finite population mean using double sampling under probability proportional to size sampling in the presence of extreme values, 2023, 9, 24058440, e21418, 10.1016/j.heliyon.2023.e21418
5.	Muhammad Junaid, Sadaf Manzoor, Sardar Hussain, M.E. Bakr, Oluwafemi Samson Balogun, Shahab Rasheed, An optimal estimation approach in non-response under simple random sampling utilizing dual auxiliary variable for finite distribution function, 2024, 10, 24058440, e38343, 10.1016/j.heliyon.2024.e38343
6.	Khazan Sher, Muhammad Ameeq, Muhammad Muneeb Hassan, Olayan Albalawi, Ayesha Afzal, Development of improved estimators of finite population mean in simple random sampling with dual auxiliaries and its application to real world problems, 2024, 10, 24058440, e30991, 10.1016/j.heliyon.2024.e30991
7.	Abdullah Mohammed Alomair, Soofia Iftikhar, Calibrated EWMA estimators for time-scaled surveys with diverse applications, 2024, 10, 24058440, e31030, 10.1016/j.heliyon.2024.e31030

Reader Comments

Your name:*

Email:*
© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)