two column

chapter8.Rmd

# Multivariate Statisttics 7.Week Lecture Notes

# Multivariate Multiple Regression Analysis

You can find this topic in the chapter 8 in the book.

1. Multiple Regression

$$  Y = X\beta +\varepsilon ~~\to \text{Regression Model} $$
$$\varepsilon \to Error \\ ~~~~~~~X \to \text{data array } \\~~~ Y \to \text{Response}$$



$$ \hat{\beta} = (X^{'}X)^{-1}X^{'}Y ~~\to \text{Estimated regressions coefficients}  $$

$$ \hat{Y} = X\hat{\beta} \to \text{Predicted response values} $$

## Example Application of Regression Analysis

```{r}
y<- c(1,4,3,8,9,4,6,3,7)
X <- c(rep(1,9),0,1,3,4,2,3,1,5,4)
X <- matrix(X, nrow= 9)
```


```{r}
X
```


```{r}
beta_hat <- solve(t(X)%*%X)%*%t(X)%*%y
beta_hat
```
The estimated regression model is $ Yhat= 3.965 + X\cdot 0.405 $

```{r}
prediction <- X%*%beta_hat
predictions <- data.frame(y=y,prediction=prediction)
predictions
```

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<div class="col2">


Multiple Regression(çoklu)



$Y_1=\beta_0+\beta_1Z_{11}+..+\beta_1Z_{1r}Z_{1r}+\varepsilon_1$


$Y_2=\beta_0+\beta_1Z_{21}+..+\beta_1Z_{1r}Z_{2r}+\varepsilon_2$


$\vdots$



$Y_n=\beta_0+\beta_1Z_{n1}+..+\beta_1Z_{1r}Z_{nr}+\varepsilon_n$



Response~ Multiple_exp_variable

$\vdots$


$\vdots$


```

```


Multivarite(Multiple)Regression(Çok değişkenli Çoklu Regresyon)

$Y_1=\beta_{01}+\beta_{11}Z_{1}+...+\beta_{r1}Z_{r}+\varepsilon_1$


$Y_2=\beta_{02}+\beta_{12}Z_{1}+...+\beta_{r2}Z_{r}+\varepsilon_2$


$\vdots$


$Y_n =\beta_{0n}+\beta_{1n}Z_{1}+...+\beta_{rn}Z_{r}+\varepsilon_n$

M response variables~explanotary variables(#1)

The error term  $\varepsilon'=[\varepsilon_1,\varepsilon_2,...,\varepsilon_m]$ has $E(\varepsilon)=\mathbf{0}$ and $var(\varepsilon)=\Sigma$ Thus ,the terms associated with different responses may be 


```

```
</div>
Explanort variables matrix
$$Z_{nx(n+1)}=\begin{bmatrix}
Z_{10} & Z_{11} & \dots &Z_{1r}\\
Z_{20} & Z_{21} & \dots &Z_{2r}\\
\vdots & \dots & \dots &\dots\\
Z_{n0} & Z_{n1} & \dots &Z_{nr}\\
\end{bmatrix}_{nx(r+1)} \\$$


Responses matrix

$$\_{nxm}=\begin{bmatrix}
Y_{11} & Y_{11} & \dots &Y_{1m}\\
Z_{21} & Z_{21} & \dots &Z_{2m}\\
\vdots & \dots & \dots &\dots\\
Z_{n1} & Z_{n2} & \dots &Z_{nm}\\
\end{bmatrix}_{nxm} \\$$

Epsilon(errors)


$$\epsilon=\begin{bmatrix}
\epsilon_{11} & \epsilon_{11} & \dots &\epsilon_{1m}\\
\epsilon_{21} & \epsilon_{21} & \dots &\epsilon_{2m}\\
\vdots & \dots & \dots &\dots\\
\epsilon_{n1} & \epsilon_{n2} & \dots &\epsilon_{nm}\\
\end{bmatrix}_{nxm} \\$$





$$\beta=\begin{bmatrix}
\beta_{01} & \beta{01} & \dots &\beta_{0m}\\
\beta_{11} & \beta_{12} & \dots &\beta_{1m}\\
\vdots & \dots & \dots &\dots\\
\beta_{r1} & \beta_{r2} & \dots &\beta_{rm}\\
\end{bmatrix}_{rxm} \\$$


$$\beta =[\beta_{1},\beta_{2},...,\beta_{n}]$$


Regression models

$$\mathbf(Y)_{nxm}=\mathbf(Z)_{nx(n+1)}+\mathbf(\beta)_{rxm}+ \mathbf(\epsilon)_{nxm}$$ 


$$ betahats=(Z'Z)^-1Z'Y_{i} $$


yhat

$$ Yhats=Z betahat $$
infect you fit m different regression models

$$Y_{İ}=ZBİ+epsiloni , i=1,...,m $$


input the datasets


```{r}
Y1=c(32,47,54,60,45,27,38,46) #1ST response variable
Y2=c(27,41,40,52,30,18,21,18)  #2ST response variable

x1=c(4,8,6,5,9,6,11,10) #1ST explonatory variable
X2=c(13,18,17,20,14,8,9,7)  #2ST explonatory variable

```

organize Y and Z
```{r}
Y1=c(32,47,54,60,45,27,38,46) #1ST response variable
Y2=c(27,41,40,52,30,18,21,18)  #2ST response variable

x1=c(4,8,6,5,9,6,11,10) #1ST explonatory variable
x2=c(13,18,17,20,14,8,9,7)  #2ST explonatory variable

```

find bhat six beta coeffiencet

```{r}
(Y<-matrix(c(Y1,Y2),nrow = 8,ncol=2))

(Z<-matrix(c(rep(1,8) ,x1,x2),nrow = 8,ncol=3))

(bhat<-solve(t(Z) %*% Z)%*% t(Z) %*% Y)

```


our models  here

$$Y_1hat=2.6995+1.8780x1+2.043x2$$

$$Y_2hat=-3.3458+0.1295x1+2.043x2$$

yhats are ztimesbhat

ozge.Rmd

Explanort variables matrix
$$Z_{nx(n+1)}=\begin{bmatrix}
Z_{10} & Z_{11} & \dots &Z_{1r}\\
Z_{20} & Z_{21} & \dots &Z_{2r}\\
\vdots & \dots & \dots &\dots\\
Z_{n0} & Z_{n1} & \dots &Z_{nr}\\
\end{bmatrix}_{nx(r+1)} \\$$


Responses matrix

$$\_{nxm}=\begin{bmatrix}
Y_{11} & Y_{11} & \dots &Y_{1m}\\
Z_{21} & Z_{21} & \dots &Z_{2m}\\
\vdots & \dots & \dots &\dots\\
Z_{n1} & Z_{n2} & \dots &Z_{nm}\\
\end{bmatrix}_{nxm} \\$$

Epsilon(errors)


$$\epsilon=\begin{bmatrix}
\epsilon_{11} & \epsilon_{11} & \dots &\epsilon_{1m}\\
\epsilon_{21} & \epsilon_{21} & \dots &\epsilon_{2m}\\
\vdots & \dots & \dots &\dots\\
\epsilon_{n1} & \epsilon_{n2} & \dots &\epsilon_{nm}\\
\end{bmatrix}_{nxm} \\$$





$$\beta=\begin{bmatrix}
\beta_{01} & \beta{01} & \dots &\beta_{0m}\\
\beta_{11} & \beta_{12} & \dots &\beta_{1m}\\
\vdots & \dots & \dots &\dots\\
\beta_{r1} & \beta_{r2} & \dots &\beta_{rm}\\
\end{bmatrix}_{rxm} \\$$


$$\beta =[\beta_{1},\beta_{2},...,\beta_{n}]$$


Regression models

$$\mathbf(Y)_{nxm}=\mathbf(Z)_{nx(n+1)}+\mathbf(\beta)_{rxm}+ \mathbf(\epsilon)_{nxm}$$ 


betahats

twocol.Rmd

<style>
  .col2 {
    columns: 2 200px;         /* number of columns and width in pixels*/
    -webkit-columns: 2 200px; /* chrome, safari */
    -moz-columns: 2 200px;    /* firefox */
  }

</style>

<div class="col2">
  firtst
$Y_1=\beta_0+\beta_1Z_{11}+..+\beta_1Z_{1r}Z_{1r}+\varepsilon
```

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  second column
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```
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