Regression analysis is a statistical technique for estimating the relationship among variables which have reason and result relation. Main focus of univariate regression is analyses the relationship between a dependent variables X 1 , ⋯ , X n and one independent variable Y and formulates the linear relation equation between dependent and independent variable.

The simple linear regression model is the simplest regression model in which we have only one predictor X.

This model, which is common in practice, is written as

Y i = b + a X i + ε i ,     i = 1 , ⋯ , n ,

where

- Y i , X i are the values of the response and predictor variables in the trial, respectively;

- The unknown parameters: a is called the intercept, and b is the slope of the line;

- ε i is usually assumed to be iid (error) from N ( 0 , σ ε 2 ) specially for inference purposes (see for instance [5] ).

Then estimates of simple linear model’s parameters should be obtained accordingly, using some method like the ordinary least squares method, which relies on minimizing the sum of square of errors ∑ ​     ε i 2 .

For the simple linear regression model the ordinary least squares estimations of a and b are

b ^ = Y ¯ − a ^ X ¯

and

a ^ = ∑ i ( X i − X ¯ ) ( Y i − Y ¯ ) / ∑ i ( X i − X ¯ ) 2

where X ¯ and Y ¯ are the mean of the variable X and the variable Y respectively.

The goodness R 2 of fit is defined as

R 2 = ∑ i ( Y i − Y ¯ ) 2 / ∑ i ( X i − X ¯ ) 2 .

Regression models with one dependent variable and more than one independent variable are called multi-linear regression (see for instance [6] ).

Multivariate regression analysis model is formulated as in the following:

Y = α 0 + α 1 X 1 + ⋯ + α n X n + ε

where

- Y is the dependent variable,

- X i is the independent variables,

- α i is the parameters,

- ε is the error.

The assumptions of multi-linear regression analysis are normal distribution, linearity, freedom extreme values and having no multiple ties between independent variables [6] .

The linear model can be written as

Y = X α + ε

where

- Y ∈ R n is the vector of observations on the dependent variable,

Y = ( Y 1 , ⋯ , Y n ) T .

- X ∈ R n × R p + 1 is the matrix consisting of a column of ones and p column vectors of the observations on the independent variables, the form of X is

1 X 11 ⋯ X 1 p ⋮ ⋮ ⋱ ⋮ 1 X n 1 ⋯ X n p

- α ∈ R p + 1 is the vector of parameters to be estimated,

α = ( α 0 , α 1 , ⋯ , α p ) T .

- ϵ ∈ R n is the vector of random errors,

ϵ = ( ϵ 0 , ϵ 1 , ⋯ , ϵ n ) T .

The vector α is a vector of unknown constants to be estimated from the data by α ^ .

The normal equations [7] are written as

X T X α ^ = X T Y .

If X T X has an inverse, then the unique solution of normal equations given by

α ^ = ( X T X ) − 1 ( X T Y ) .

The vector Y ^ of estimated means of the dependent variable Y for the values of the independent variables X 1 , ⋯ , X n in the dataset is computed as

Y ^ = X α ^ .

However, to express Y ^ as a linear function of Y. Thus,

Y ^ = [ X ( X T X ) − 1 X T ] Y .

In this section we conceder the following integer variables:

- X 1 the frequency of plural present verbs with a form (y―un, ي---ون) or there inverse (ly―un, لي---ون);

- X 2 the frequency of plural present verbs with a form (t―un, ت---ون) or there inverse (lt―un, لت---ون);

- X 3 the frequency of the verbs in the above form without (t,ت) or (y,ي ).

We use the software of Quran statistics [8] let Y = X 1 + X 2 + X 3 for determined the set of triple ( Y i , X 1 i , X 2 i ) , i = 1 , ⋯ , 387 .

Let Y 7 , Y 9 , Y 32 is the sum of frequency of all above derivative verbs with same form.

However, the dataset are given in Table 1.

Since X 3 are not a frequency of plural present verbs [9] then, we let X = X 1 + X 2 a dependent variable and Y an independent variable for simple linear regression model. However, the simple linear regression model of verbs is:

Y ≈ a X + b ,

where a and b are the parameters of the model.

For estimate the parameters a and b, calculate the coefficient of correlation R, plotting a fit and test hypothesis of simple linear regression model we used the following matlab codes:

[r, m, b] = regression (X, Y);

plotregression (X, Y);

fitlm (X, Y);

The results of regression m function is given as the parameters a = 1 and b = 1 the coefficient of correlation R = 0.91441 , see also Figure 1. And the results of test hypotheses by fitlm. m function are:

( Y 86 , X 86 ) is frequency of verbs of enter (yadkolun دخلون ، يدخلون dakilun).

Also there exist 17 point in dataset in the line fit their equation is Y = X + 1 , when we use the commend find ( Y = X + 1 ) there indexes in dataset are:

For more information’s about the correspondence of this points in the dataset see Table 2 see for instance [10] [11] .3.3. Multi Linear Regression Model of VerbsFor multi linear regression model of verbs: Y ≈ α 1 X 1 + α 2 X 2 + α 3 where α 1 , α 2 and α 3 are the parameters of the model.For estimate the parameters α 1 , α 2 and α 3 , and test hypothesis of multi linear regression model we used the following matlab code:

Number of observations: 387, Error degrees of freedom: 384Root Mean Squared Error: 6.18R-squared: 0.836, Adjusted R-Squared 0.835F-statistic vs. constant model: 981, p-value = 1.2e−151Also in the Figure 2, we show the name of Allah in Arabic الله.