**4. General linear hypothesis**

In this section, we consider to test a general linear hypothesis

$$H\_{\boldsymbol{\beta}}: \mathbf{C} \boldsymbol{\Theta} \mathbf{D} = \mathbf{O},\tag{4.1}$$

against alternatives *Kg* : **C***Θ***D** ≠ **O** under a multivariate linear model given by (2.1), where **C** is a *c* × *k* given matrix with rank *c* and **D** is a *p* × *d* given matrix with rank *d*. When **C** = (**O I***k* − *j* ) and **D** = **I***p*, the hypothesis *Hg* becomes *H* : *Θ*2 = **O**.

For the derivation of LR test of (4.1), we can use the following conventional approach: If **U**=**YD,** then the rows of **U** are independent and normally distributed with the identical covariance matrix **D**′*Σ***D**, and

$$\mathbf{E}(\mathbf{U}) = \mathbf{X}\mathbf{E},\tag{4.2}$$

where **Ξ**=**ΘD**. The hypothesis (4.1) is expressed as

$$H\_g: \mathbf{C}\Xi = \mathbf{O}.\tag{4.3}$$

Applying a general theory for testing *Hg* in (2.1), we have the LRC *λ*:

$$\mathcal{X}^{2/n} = \Lambda = \frac{|\mathbf{S}\_e|}{|\mathbf{S}\_e + \mathbf{S}\_h|},\tag{4.4}$$

where

$$\begin{aligned} \mathbf{S}\_{\ast} &= \mathbf{U} (\mathbf{I}\_{\ast} - \mathbf{P}\_{\mathbf{x}}) \mathbf{U} \\ &= \mathbf{D}' \mathbf{V}' (\mathbf{I}\_{\ast} - \mathbf{P}\_{\mathbf{A}}) \mathbf{V} \mathbf{D}\_{\ast} \end{aligned}$$

and

$$\begin{aligned} \mathbf{S}\_{\mathbb{A}} &= (\mathbf{C}(\mathbf{X}\mathbf{X})^{-1}\mathbf{X}\mathbf{U})^{\prime}\langle\mathbf{C}(\mathbf{X}\mathbf{X})^{-1}\mathbf{C}\rangle^{-1}\mathbf{C}(\mathbf{X}\mathbf{X})^{-1}\mathbf{X}\mathbf{U}, \\ &= (\mathbf{C}(\mathbf{X}^{\prime}\mathbf{X})^{-1}\mathbf{X}^{\prime}\mathbf{V}\mathbf{D})^{\prime}\langle\mathbf{C}(\mathbf{X}^{\prime}\mathbf{X})^{-1}\mathbf{C}\rangle^{-1}\mathbf{C}(\mathbf{X}\mathbf{X})^{-1}\mathbf{X}^{\prime}\mathbf{V}\mathbf{D}. \end{aligned}$$

**Theorem 4.1***The statistic Λ in* (4.4) *is an LR statistic for testing* (4.1) *under* (2.1)*. Further, under Hg, Λ* ∼ *Λd*(*c*, *n* − *k*)*.*

*Proof.* Let **G** = (**G**1 **G**2) be a *p* × *p* matrix such that **G**1 = **D, G**1 **′ G**2 =**O,** and |**G**| ≠ 0. Consider a transformation from **Y** to (**U V**) =**Y**(**G**1 **G**2).

Then the rows of (**U V**) are independently normal with the same covariance matrix

$$\Psi = \mathbf{G} \mathfrak{D} \mathbf{G} = \begin{pmatrix} \Psi\_{11} & \Psi\_{12} \\ \Psi\_{21} & \Psi\_{22} \end{pmatrix}, \quad \Psi\_{12} : d \times (p - d),$$

and

The expressions (2.11) for **S***e* and **S***h* in terms of **S** can be obtained from projection matrices

W +-

RRR

ÇW - -

R

= [ ] = [ ] [( )X], = [( ) ].

**X 1 IP IP P X**

*n n n n*

against alternatives *Kg* : **C***Θ***D** ≠ **O** under a multivariate linear model given by (2.1), where **C** is a *c* × *k* given matrix with rank *c* and **D** is a *p* × *d* given matrix with rank *d*. When **C** = (**O**

For the derivation of LR test of (4.1), we can use the following conventional approach: If **U**=**YD,** then the rows of **U** are independent and normally distributed with the identical covariance

w^

In this section, we consider to test a general linear hypothesis

) and **D** = **I***p*, the hypothesis *Hg* becomes *H* : *Θ*2 = **O**.

where **Ξ**=**ΘD**. The hypothesis (4.1) is expressed as

Applying a general theory for testing *Hg* in (2.1), we have the LRC *λ*:

lL

2/ | | == , | | *n e*

> = '( ) = ( ),

**SU U**

**I P DIPD**

*n* - ¢ ¢ - **X A**

**Y Y**

*e n*

*e h*

+ **S**

**4. General linear hypothesis**

( )2 1

**1**

: =, *Hg* **CD O** Q (4.1)

E( ) = , **U X**X (4.2)

: =. *Hg* **C O** X (4.3)

**S S** (4.4)

*n n n*

**1 IP X 1**

based on

148 Applied Linear Algebra in Action

**I***k* − *j*

where

and

matrix **D**′*Σ***D**, and

$$\begin{aligned} E[(\mathbf{U}\,\mathbf{V})] &= \mathbf{X}\Theta(\mathbf{G}\_1\,\mathbf{G}\_2) \\ &= \mathbf{X}(\Xi\,\Lambda), \quad \Xi = \Theta\mathbf{G}\_1, \,\Lambda = \Theta\mathbf{G}\_2. \end{aligned}$$

The conditional of **V** given **U** is normal. The rows of **V** given **U** are independently normal with the same covariance matrix Ψ11⋅2 , and

$$\begin{aligned} \mathbf{E}(\mathbf{V} \mid \mathbf{U}) &= \mathbf{X}\mathbf{A} + (\mathbf{U} - \mathbf{X}\mathbf{E})\mathbf{T}, \\ &= \mathbf{X}\mathbf{A}^\* + \mathbf{U}\Gamma, \end{aligned}$$

where **Δ**\* =**Δ**−**ΞΓ** and **Γ**=**Ψ**11 −1**Ψ**12. We see that the maximum likelihood of **V** given **U** does not depend on the hypothesis. Therefore, an LR statistic is obtained from the marginal distribution of **U,** which implies the results required.

#### **5. Additional information tests for response variables**

We consider a multivariate regression model with an intercept term *x*0 and *k* explanatory variables *x*1, …, *xk* as follows.

$$\mathbf{Y} = \mathbf{1}\mathbf{\bar{O}}' + \mathbf{X}\Theta + \mathbf{E},\tag{5.1}$$

where **Y** and **X** are the observation matrices on *y* = (*y*1, …, *yp*)′ and *x* = (*x*1, …, *xk*)′. We assume that the error matrix **E** has the same property as in (2.1), and rank (**1***n* **X**) = *k* + 1. Our interest is to test a hypothesis *H*2 ⋅ 1 on no additional information of *y*2 = (*yq* + 1, …, *yp*)′ in presence of *y*1 = (*y*1, …, *yq*)′.

Along the partition of *y* into (*y*1′, *y*2′) let **Y,** *θ,* Θ, and Σ partition as

$$\begin{array}{cc} \mathbf{M} = (\mathbf{M}\_1 \; \mathbf{M}\_2), & \mathbf{O} = (\mathbf{O}\_1 \; \mathbf{O}\_2), \\ \mathbf{H} = \begin{pmatrix} \mathbf{O}\_1 \\ \mathbf{O}\_2 \end{pmatrix}, & \mathbf{E} = \begin{pmatrix} \mathbf{E}\_{11} & \mathbf{E}\_{12} \\ \mathbf{E}\_{21} & \mathbf{E}\_{22} \end{pmatrix}. \end{array}$$

The conditional distribution of **Y**2 given **Y**1 is normal with mean

$$\begin{split} \mathbb{E}(\mathbf{Y}\_{2} \mid \mathbf{Y}\_{1}) &= \mathbf{1}\boldsymbol{\theta}\_{2}^{\prime} + \mathbf{X}\boldsymbol{\Theta}\_{2} + (\mathbf{Y}\_{1} - \mathbf{1}\_{n}\boldsymbol{\theta}\_{1}^{\prime} - \mathbf{X}\boldsymbol{\Theta}\_{1})\boldsymbol{\Sigma}\_{11}^{-1}\boldsymbol{\Sigma}\_{12} \\ &= \mathbf{1}\_{n}\tilde{\boldsymbol{\theta}}\_{02}^{\prime} + \mathbf{X}\tilde{\boldsymbol{\Theta}}\_{2} + \mathbf{Y}\_{1}\mathbf{Z}\_{11}^{-1}\boldsymbol{\Sigma}\_{12}, \end{split} \tag{5.2}$$

and the conditional covariance matrix is expressed as

$$\text{Var}[\text{VecC}\left(\mathbf{Y}\_2 \mid \mathbf{Y}\_1\right)] = \mathbf{Z}\_{22\cdot l} \otimes \mathbf{I}\_u,\tag{5.3}$$

where **Σ**22⋅1 =**Σ**22 −**Σ**21**Σ**11 −1**Σ**12 , and

$$
\tilde{\boldsymbol{\Theta}}'\_2 = \boldsymbol{\Theta}'\_2 - \boldsymbol{\Theta}'\_1 \boldsymbol{\Sigma}\_{11}^{-1} \boldsymbol{\Sigma}\_{12}, \quad \tilde{\boldsymbol{\Theta}}\_2 = \boldsymbol{\Theta}\_2 - \boldsymbol{\Theta}\_1 \boldsymbol{\Sigma}\_{11}^{-1} \boldsymbol{\Sigma}\_{12}.
$$

Here, for an *n* × *p* matrix **Y**=(**y**(1), …, **y**(p,)) v*ec* (**Y**) means an *np*-vector (**y**(1) *′* , …, **y**( *p*) *′* )′ . Now we define the hypothesis *H*2 ⋅ 1 as

$$H\_{2,1} \colon \Theta\_2 = \Theta\_1 \Sigma\_{11}^{-1} \Sigma\_{12} \Longleftrightarrow \tilde{\Theta}\_2 = \mathcal{O}.\tag{5.4}$$

The hypothesis *H*2 ⋅ 1 means that *y*2 after removing the effects of *y*1 does not depend on *x*. In other words, the relationship between *y*2 and *x* can be described by the relationship between *y*1 and **x**. In this sense, **y**2 is redundant in the relationship between *y* and *x*.

The LR criterion for testing the hypothesis *H*2 ⋅1 against alternatives *K*2⋅1 :**Θ**˜ 2⋅1 ≠**O** can be obtained through the following steps.

(D1) The density function of **Y**=(**Y**1 **Y**2) can be expressed as the product of the marginal density function of **Y**1 and the conditional density function of **Y**2 given **Y**1. Note that the density functions of **Y**1 under *H*2 ⋅ 1 and *K*2 ⋅ 1 are the same.

(D2) The spaces spanned by each column of E(**Y**2 |**Y**1) are the same, and let the spaces under *K*2 ⋅ 1 and *H*2 ⋅ 1 denote by *Ω* and *ω,* respectively. Then

$$
\Omega = \mathcal{R}[(\mathbf{1}\_n \ \mathbf{Y}\_1 \ \mathbf{X})], \quad \boldsymbol{\alpha} = \mathcal{R}[(\mathbf{1}\_n \ \mathbf{Y}\_1)],
$$

and dim(*Ω*) = *q* + *k* + 1, dim(*ω*) = *k* + 1.

to test a hypothesis *H*2 ⋅ 1 on no additional information of *y*2 = (*yq* + 1, …, *yp*)′ in presence of

1 2 1 2 1 11 12 2 21 22

Q Q Q S S

S S

1

**Y** (5.2)

*′* , …, **y**( *p*) *′* )′

. Now we

2⋅1 ≠**O** can be

Var[ ( | )] = , vec 2 1 22 1× Ä *n* **Y Y** S **I** (5.3)

2 1 2 1 11 12 2 : = = O. - *H* × Q Q S S Û Q% (5.4)

Q S S

= ,= . æö æ ö ç÷ ç ÷ èø è ø

S

2 1 2 2 1 1 1 11 12 1

Q S S

¢ ¢ + Q+ - -

**1X 1 X**

*n*

q

1 1 2 2 1 11 12 2 2 1 11 12 = ,= . - - ¢¢¢ - -

S S Q% Q Q S S

02 2 1 11 12

q

**Y YY**

q

E( | ) = ( ) = ,

¢ + +

**1 X** % %

q

Here, for an *n* × *p* matrix **Y**=(**y**(1), …, **y**(p,)) v*ec* (**Y**) means an *np*-vector (**y**(1)

*y*1 and **x**. In this sense, **y**2 is redundant in the relationship between *y* and *x*.

The LR criterion for testing the hypothesis *H*2 ⋅1 against alternatives *K*2⋅1 :**Θ**˜

1

The hypothesis *H*2 ⋅ 1 means that *y*2 after removing the effects of *y*1 does not depend on *x*. In other words, the relationship between *y*2 and *x* can be described by the relationship between

(D1) The density function of **Y**=(**Y**1 **Y**2) can be expressed as the product of the marginal density function of **Y**1 and the conditional density function of **Y**2 given **Y**1. Note that the density

= ( ), = ( ),

Along the partition of *y* into (*y*1′, *y*2′) let **Y,** *θ,* Θ, and Σ partition as

q

The conditional distribution of **Y**2 given **Y**1 is normal with mean

*n*

and the conditional covariance matrix is expressed as

−1**Σ**12 , and

q

% q

q

**Y Y Y**

q

*y*1 = (*y*1, …, *yq*)′.

150 Applied Linear Algebra in Action

where **Σ**22⋅1 =**Σ**22 −**Σ**21**Σ**11

define the hypothesis *H*2 ⋅ 1 as

obtained through the following steps.

functions of **Y**1 under *H*2 ⋅ 1 and *K*2 ⋅ 1 are the same.

(D3) The likelihood ratio criterion *λ* is expressed as

$$\mathcal{A}^{2/n} = \Lambda = \frac{|\mathbf{S}\_{\Omega}|}{|\mathbf{S}\_{\alpha}|} = \frac{|\mathbf{S}\_{\Omega}|}{|\mathbf{S}\_{\Omega} + (\mathbf{S}\_{\alpha} - \mathbf{S}\_{\Omega})|}.$$

where **S***Ω* =**Y**2 **′** (**I***n* −**P***Ω*)**Y**2 and **S***ω* =**Y**2 **′** (**I***n* −**P***ω*)**Y**2.

(D4) Note that E(**Y**2 |**Y**1)′ (**P***ω* −**P***ω*)E(**Y**2 |**Y**1) =**O** under *H*2 ⋅ 1. The conditional distribution of *Λ* under *H*2 ⋅ 1 is *Λp* − *q*(*k*, *n* − *q* − *k* − 1), and hence the distribution of *Λ* under *H*2 ⋅ 1 is *Λp* − *q*(*k*, *n* − *q* − *k* − 1).

Note that the *Λ* statistic is defined through **Y**2 **′** (**I***n* −**P***Ω*)**Y**2 and **Y**2 **′** (**P***Ω* −**P***ω*)**Y**2 , which involve *n* × *n* matrices. We try to write these statistics in terms of the SSP matrix of (*y*′, *x*′)′ defined by

$$\begin{aligned} \mathbf{S} &= \sum\_{i=1}^{n} \begin{pmatrix} \mathbf{y}\_{i} - \overline{\mathbf{y}} \\ \mathbf{x}\_{i} - \overline{\mathbf{x}} \end{pmatrix} \begin{pmatrix} \mathbf{y}\_{i} - \overline{\mathbf{y}} \\ \mathbf{x}\_{i} - \overline{\mathbf{x}} \end{pmatrix}, \\ &= \begin{pmatrix} \mathbf{S}\_{\overline{\mathbf{y}}} & \mathbf{S}\_{\overline{\mathbf{x}}} \\ \mathbf{S}\_{\overline{\mathbf{y}}} & \mathbf{S}\_{\overline{\mathbf{x}}} \end{pmatrix}, \end{aligned}$$

where **y**¯ and **x**¯ are the sample mean vectors. Along the partition of **y**=(**y**1 *′* , **y**2 *′* )′, we partition **S** as

$$\mathbf{S} = \begin{pmatrix} \mathbf{S}\_{11} & \mathbf{S}\_{12} & \mathbf{S}\_{1\times} \\ \mathbf{S}\_{21} & \mathbf{S}\_{22} & \mathbf{S}\_{2\times} \\ \mathbf{S}\_{\times 1} & \mathbf{S}\_{\times 2} & \mathbf{S}\_{\times \times} \end{pmatrix}.$$

We can show that

$$\begin{aligned} \mathbf{S}\_{\boldsymbol{\alpha}} &= \mathbf{S}\_{22 \cdot 1} = \mathbf{S}\_{22} - \mathbf{S}\_{21} \mathbf{S}\_{11}^{-1} \mathbf{S}\_{12}, \\ \mathbf{S}\_{\boldsymbol{\alpha}} &= \mathbf{S}\_{22 \cdot 1 \times} = \mathbf{S}\_{22 \cdot \boldsymbol{\alpha}} - \mathbf{S}\_{21 \cdot \boldsymbol{\alpha}} \mathbf{S}\_{11 \cdot \boldsymbol{\alpha}}^{-1} \mathbf{S}\_{12 \cdot \boldsymbol{\alpha}}. \end{aligned}$$

The first result is obtained by using

$$
\rho = \mathcal{R}[\mathbf{1}\_n] + \mathcal{R}[(\mathbf{I}\_n - \mathbf{P}\_{\mathbf{1}\_n})\mathbf{Y}\_1].
$$

The second result is obtained by using

$$\begin{split} \boldsymbol{\Omega} &= \mathcal{R}[\mathbf{1}\_{\boldsymbol{\pi}}] + \mathcal{R}[(\boldsymbol{\tilde{\mathbf{Y}}}\_{\mathrm{l}}, \boldsymbol{\tilde{\mathbf{X}}})] \\ &= \mathcal{R}[\mathbf{1}\_{\boldsymbol{\pi}}] + \mathcal{R}[(\mathbf{I}\_{\boldsymbol{\pi}} - \mathbf{P}\_{0})\mathbf{X}] + \mathcal{R}[(\mathbf{I}\_{\boldsymbol{\pi}} - \mathbf{P}\_{\mathbf{X}})(\mathbf{I}\_{\boldsymbol{\pi}} - \mathbf{P}\_{0})\mathbf{Y}\_{\boldsymbol{\pi}}], \end{split}$$

where **Y**˜ 1 =(**I***n* −**P1***n* )**Y**1 and **X**˜ =(**I***n* −**P1***n* )**X**.

Summarizing the above results, we have the following theorem.

**Theorem 5.1***In the multivariate regression model* (5.1)*, consider to test the hypothesis H*2 ⋅1 *in* (5.4) *against K*2 ⋅ 1*. Then the LR criterion λ is given by*

$$\mathcal{A}^{2\vee n} = \Lambda = \frac{|\mathbf{S}\_{22\cdot \mathbb{N}}|}{|\mathbf{S}\_{22\cdot \mathbb{I}}|},$$

*whose null distribution isΛp* − *q*(*k*, *n* − *q* − *k* − 1).

Note that **S**22⋅1 can be decomposed as

$$\mathbf{S}\_{22\cdot 1} = \mathbf{S}\_{22\cdot 1\times} + \mathbf{S}\_{2\times 1} \mathbf{S}\_{\times \times 1}^{-1} \mathbf{S}\_{\times 2\cdot 1}.$$

This decomposition is obtained by expressing **S**22⋅1*x* in terms of **S**22⋅1 , **S**2*x*⋅1 , **S***xx*⋅1 , and **S***x*2⋅1 by using an inverse formula

$$
\begin{pmatrix}
\mathbf{H}\_{11} & \mathbf{H}\_{12} \\
\mathbf{H}\_{21} & \mathbf{H}\_{22}
\end{pmatrix}^{-1} = \begin{pmatrix}
\mathbf{H}\_{11}^{-1} & \mathbf{O} \\
\mathbf{O} & \mathbf{O}
\end{pmatrix} + \begin{pmatrix}
\mathbf{I}
\end{pmatrix} \mathbf{H}\_{22,1}^{-1} \begin{pmatrix}
\end{pmatrix}.\tag{5.5}
$$

The decomposition is expressed as

$$\mathbf{S}\_{22\cdot 1} - \mathbf{S}\_{22\cdot 1x} = \mathbf{S}\_{2x\cdot 1} \mathbf{S}\_{x\cdot 1}^{-1} \mathbf{S}\_{x2\cdot 1}.\tag{5.6}$$

The result may be also obtained by the following algebraic method. We have

$$\begin{aligned} \mathbf{S}\_{22 \cdot 1} - \mathbf{S}\_{22 \cdot 1x} &= \mathbf{Y}\_2' (\mathbf{P}\_{\Omega} - \mathbf{P}\_{\boldsymbol{\alpha}}) \mathbf{Y}\_2 \\ &= \mathbf{Y}\_2' (\mathbf{P}\_{\boldsymbol{\alpha}^\perp \cap \Omega}) \mathbf{Y}\_2, \end{aligned}$$

and

= [ ] [( ) ]. *n n* 1 *n*

= [ ] [( ) ] [( )( ) ],

**1 I P I PI P**

*n n n n* **X**

**Theorem 5.1***In the multivariate regression model* (5.1)*, consider to test the hypothesis H*2 ⋅1 *in* (5.4)

2/ 22 1

L **S S**

22 1 22 1 2 1 1 2 1 = . *x x xx x* - **S S SSS** × × ×× × +

This decomposition is obtained by expressing **S**22⋅1*x* in terms of **S**22⋅1 , **S**2*x*⋅1 , **S***xx*⋅1 , and **S***x*2⋅1

( ) 1 1 1

= .

1

22 1 22 1 2 2

**SS Y Y**

×× W

2 2

w

^w ÇW

**Y Y**

=( ) = ( ),

**P**

11 12 11 11 12 1 1

**H H H O HH H HH I**

22 1 22 1 2 1 1 2 1 = . *x x xx x*

æö - æ öæ ö ç ÷ ç ÷ç ÷ + è ø è øè ø

The result may be also obtained by the following algebraic method. We have

×

22 1 | | == , | | *n x*

×

1

22 1 21 11

**H H OO I** (5.5)

+-+- -

0 0 1

**X Y**

+ - **1** R R **1 I PY**

w

= [ ] [( , )]

% % *n*

W +

)**Y**1 and **X**˜ =(**I***n* −**P1***n*

*against K*2 ⋅ 1*. Then the LR criterion λ is given by*

*whose null distribution isΛp* − *q*(*k*, *n* − *q* − *k* − 1).

21 22

The decomposition is expressed as

Note that **S**22⋅1 can be decomposed as

by using an inverse formula

**1**

R R

Summarizing the above results, we have the following theorem.

1

RR R **Y X**

l

)**X**.

The second result is obtained by using

where **Y**˜ 1 =(**I***n* −**P1***n*

152 Applied Linear Algebra in Action

$$
\Omega = \mathcal{R}[\mathbf{1}\_{\boldsymbol{u}}] + \mathcal{R}[(\tilde{\mathbf{Y}}\_{\boldsymbol{1}} \tilde{\mathbf{X}})], \quad \boldsymbol{\phi} = \mathcal{R}[\mathbf{1}\_{\boldsymbol{u}}] + \mathcal{R}[\tilde{\mathbf{Y}}\_{\boldsymbol{1}}].
$$

Therefore,

$$\begin{aligned} \boldsymbol{\alpha}^{\perp} \cap \Omega &= \mathcal{R} [ (\mathbf{I}\_{\boldsymbol{\pi}} - \mathbf{P}\_{\mathbf{1}\_{\boldsymbol{\pi}}} - \mathbf{P}\_{\boldsymbol{\tilde{\mathbf{y}}}\_{\boldsymbol{\tilde{\mathbf{y}}}}}) (\mathbf{\tilde{\mathbf{Y}}}\_{\boldsymbol{1}} \mathbf{\tilde{X}}) ], \\ &= \mathcal{R} [ (\mathbf{I}\_{\boldsymbol{\pi}} - \mathbf{P}\_{\mathbf{1}\_{\boldsymbol{\pi}}} - \mathbf{P}\_{\boldsymbol{\tilde{\mathbf{y}}}\_{\boldsymbol{1}}}) \mathbf{\tilde{X}} ], \end{aligned}$$

which gives an expression for **P***ω* ⊥∩*Ω* by using Theorem 3.1 (1). This leads to (5.6).


**6. Biocontrol activity of yeast against epiphytic molds**

The molds were provided from the culture collection of the University of Castilla-La Mancha (UCLM) and IVICAM (Grapevine and Wine Institute of Castilla-La Mancha). They were *Phaeomoniella* (*Pa.*) *chlamydospora*, *Neofusicoccum parvum*, *Diplodia seriata*, *Phaeoacremonium* (*Pm.*) *aleophilum* and *Aspergillus niger*.

Fungi were grown in YM agar, and pieces of agar with fungal mycelium were inserted in wells excavated in the YM agar which had been previously inoculated with yeast strains.

The results showed that there were both inter- and intraspecific variabilities.

*H. meyeri*, *H. uvarum*, *H. vineae* and *H. valbyensis* scarcely controlled fungal growth, and mycelium grew as in the control except for six *H. osmophila* which showed a good action against them.

However, *P. anomalous*, *P. galeiformis* and *P. kudriavzevii* effectively controlled all fungal strains including *A. niger*. Also, all *S. cerevisiae* strains except one presented good fungal growth control behaviour towards all the molds, and *A. niger* was inhibited effectively by only one of these strains. Additionally, the different *C. ethanolica* and *C. sake* have an effective action on the fungal growth, except in the case of *C. lactis-condensi*.

Finally, *T. delbrueckii* and *S'codes ludwigii* strains proved to have a large biocontrol effect not only because of their action against the growth but also because they affected every mold.

Most of the yeasts grew rapidly, forming a very dense lawn after 2 days of growth, suggesting that the mechanism of control might be based on a competition for space and nutrients. To qualitatively analyse the degree of competition between yeast and mold, the 0-day test was carried out afterwards. The assay was carried out with the yeast species which presented the best result in the previous experiment (**Figure 7**), allowing the detection of a high degree of competition between the two microorganisms.

#### **Figure 7.**

*Biocontrol efficacy of yeast species with 2 days (a) and 0 days (b) of preincubation time. 3: Very effective control, 2: Effective control (fungal mycelium growing slightly beyond the plug), 1: Slight control (fungal mycelium spreading in an evident form), 0: With fungal mycelium spreading similarly to the control.*

**177**

**Figure 8.**

*Yeast from Distillery Plants: A New Approach DOI: http://dx.doi.org/10.5772/intechopen.86291*

compounds.

tested with *H. osmophila*.

the conditions tested.

*The Pichia* species and the only *S'codes ludwigii* assayed offered a high degree of control. One of the conclusions given by these trials is that the competition between yeast and mold for nutrients and space appeared from the first moment of contact, probably due to the very different growth rates, i.e. the yeasts have a high rate and rapidly colonize the medium preventing the development of molds. However, the inhibition mechanism may be associated with other antagonistic or enzymatic activities occurring via the production of some active

With the aim of verifying if the inhibition mechanism was produced by cell metabolites or cell wall components, the biocontrol assays were carried out with viable yeast cells, cell extract and filtered supernatant. To carry out the experiment, four wells were excavated at different points on growth fungal plates and were filled with each faction and a negative control (lysis buffer). All of them were incubated

In most of the tests, an inhibition halo was observed with cell extracts, but when compared to the control (lysis buffer), it was difficult to identify a clear discrimination. Nevertheless, with some cell extracts, an inhibition halo slightly larger than that of the control was observed but only related to *Pm. aleophilum*. No supernatant showed antifungal activity except *H. uvarum* against *A. niger* (**Figure 8**). Finally, whole cells inhibited the molds in most cases, which is consistent with previous results except for *A. niger* which was

On the other hand, enzymatic activity such as in pectinolytic enzymes and chitinase was studied. The tests were carried out to know if the yeasts were able to degrade polygalacturonic acid and chitin. For both activities, the presence of a hydrolysis halo around the colony was considered a positive result; nevertheless, chitinolytic and pectinolytic activities were not observed in the yeasts assayed in

*Biocontrol efficacy of whole cells, cell extracts and supernatants from yeast species. 3: Very effective control, 2: Effective control (fungal mycelium growing slightly beyond the plug), 1: Slight control (fungal mycelium*

*spreading in an evident form), 0: With fungal mycelium spreading similarly to the control.*

at 30°C for a maximum of 5 days in a wet chamber [30].

*Yeast from Distillery Plants: A New Approach DOI: http://dx.doi.org/10.5772/intechopen.86291*

*Advances in Grape and Wine Biotechnology*

affected every mold.

fungal growth control behaviour towards all the molds, and *A. niger* was inhibited effectively by only one of these strains. Additionally, the different *C. ethanolica* and *C. sake* have an effective action on the fungal growth, except in the case of *C. lactis-condensi*. Finally, *T. delbrueckii* and *S'codes ludwigii* strains proved to have a large biocontrol effect not only because of their action against the growth but also because they

Most of the yeasts grew rapidly, forming a very dense lawn after 2 days of growth, suggesting that the mechanism of control might be based on a competition for space and nutrients. To qualitatively analyse the degree of competition between yeast and mold, the 0-day test was carried out afterwards. The assay was carried out with the yeast species which presented the best result in the previous experiment (**Figure 7**), allowing the

detection of a high degree of competition between the two microorganisms.

**176**

**Figure 7.**

*Biocontrol efficacy of yeast species with 2 days (a) and 0 days (b) of preincubation time. 3: Very effective control, 2: Effective control (fungal mycelium growing slightly beyond the plug), 1: Slight control (fungal mycelium spreading in an evident form), 0: With fungal mycelium spreading similarly to the control.*

*The Pichia* species and the only *S'codes ludwigii* assayed offered a high degree of control. One of the conclusions given by these trials is that the competition between yeast and mold for nutrients and space appeared from the first moment of contact, probably due to the very different growth rates, i.e. the yeasts have a high rate and rapidly colonize the medium preventing the development of molds. However, the inhibition mechanism may be associated with other antagonistic or enzymatic activities occurring via the production of some active compounds.

With the aim of verifying if the inhibition mechanism was produced by cell metabolites or cell wall components, the biocontrol assays were carried out with viable yeast cells, cell extract and filtered supernatant. To carry out the experiment, four wells were excavated at different points on growth fungal plates and were filled with each faction and a negative control (lysis buffer). All of them were incubated at 30°C for a maximum of 5 days in a wet chamber [30].

In most of the tests, an inhibition halo was observed with cell extracts, but when compared to the control (lysis buffer), it was difficult to identify a clear discrimination. Nevertheless, with some cell extracts, an inhibition halo slightly larger than that of the control was observed but only related to *Pm. aleophilum*. No supernatant showed antifungal activity except *H. uvarum* against *A. niger* (**Figure 8**). Finally, whole cells inhibited the molds in most cases, which is consistent with previous results except for *A. niger* which was tested with *H. osmophila*.

On the other hand, enzymatic activity such as in pectinolytic enzymes and chitinase was studied. The tests were carried out to know if the yeasts were able to degrade polygalacturonic acid and chitin. For both activities, the presence of a hydrolysis halo around the colony was considered a positive result; nevertheless, chitinolytic and pectinolytic activities were not observed in the yeasts assayed in the conditions tested.

#### **Figure 8.**

*Biocontrol efficacy of whole cells, cell extracts and supernatants from yeast species. 3: Very effective control, 2: Effective control (fungal mycelium growing slightly beyond the plug), 1: Slight control (fungal mycelium spreading in an evident form), 0: With fungal mycelium spreading similarly to the control.*

## **7. Bioaccumulation of heavy metals**

For bioremediation proposals, a selective elimination of metals using yeasts combined with other processes could be a feasible strategy.

Different metallic ions were tested [Cr (VI), Pb (II), Cd (II)]. Metal solutions added to inactivate biomass (obtained by thermal treatment, 5 min/121°C) were incubated at 20°C with horizontal shaking (150 rpm). Aliquots before inoculation and at time 0, 0.2, 3, 6, 24 and 48 hours were taken.

Metallic ion determination was performed by means of an inductively coupled plasma optical emission spectrometer (ICP-OES: Varian Vista-Pro, Mulgrave, VIC, Australia). Tests were semiquantitative.

Very different results were obtained depending on the yeast species as well as the metal tested for the bioaccumulation experiment (**Table 4**). The greatest metal elimination took place for Pb (II) with *H. meyeri, Z. bailii*, *P. membranaefaciens, P. kudriavzevii* and *S'codes ludwigii*, which presented an elimination range of around 20%, reaching 30% in some cases.

This percentage diminished by nearly half for Cd (II), with *P. kudriavzevii* having produced the highest elimination, followed by *Z. fermentati*.

Cr (VI) was eliminated in a much lower proportion, highlighting only *P. membranaefaciens* with 10% elimination, followed by the majority of the yeasts in which adsorption was not detected or was very low.

In general, the metal removal was instantaneous, and during the first 10 min of contact, no additional adsorption was observed. However, in some cases, *S'codes ludwigii* for Pb (II) and *H. uvarum* for Cd (II), the adsorption was progressive, possibly due to the different compositions of polysaccharides and proteins in the cell wall [31]. Unfortunately, *S. cerevisiae*, a by-product of the wine industry and suitable for this type of process, offered a low percentage of elimination for Pb (II) and a medium percentage for the other two metals compared with the rest of the yeasts of the same group. Appreciable desorption processes were not observed, although *P. kudriavzevii* released Cr (VI) into the media after 6 h of contact.

### **8. Conclusions**

This initial study of yeast populations isolated from very old distilleries reflects the great existing biodiversity of this valuable yeast niche. This contrasts with what occurs in wine cellars, where the intra and interspecific variability of yeasts have been reduced drastically due to the starter use. *Saccharomyces*, *Pichia* and *Candida* are the genera found in large proportions. Some species were only isolated for certain substrates, like *T. delbrueckii* in sweet piquettes and *P. galeiformis* in fermented piquettes.

The yeast biota of these environments is varied, so these ecological niches are microbial reserves of undoubted biotechnological interest.

In fact, a great number of thermophilic *Saccharomyces* strains with a great cell vitality were found to have potential use as starters in distillery plants.

On the other hand, yeasts coming from very old distilleries might be used as biocontrol and bioremediation agents. *Pichia* sp. inhibited all molds effectively and might be produced in an aerated fermentation process and used as an antifungal postharvest treatment of fruits. In the case of *S'codes ludwigii*, *P. membranaefaciens* and *P. kudriavzevii*, the elimination of Pb (II) was achieved, with the adsorption being almost instantaneous.