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Some notes on Relativity and other arguments   Informazione semantica nella tassonomia di Sundevall <p style="text-align: justify;">Nel 17&deg; secolo nacque la prima tassonomia ornitologica ad opera di Ray (1678) che suddivideva gli uccelli in due semplici categorie: gli uccelli d'acqua (water bird) e gli uccelli di terra (land bird) a seconda della struttura del loro becco e delle loro zampe (vedi note precedenti).</p> <p style="text-align: justify;">All'inizio del 19&deg; secolo&nbsp;furono scoperti diversi nuovi uccelli che non riuscivano a ricadere nella tassonomia di Ray - vedi ad es. lo 'screamer' nel Sud America - e che portarono ad una ridefinizione della tassonomia su nuove basi; la nuova tassonomia fu proposta da Sundevall intorno al 1830.</p> <p style="text-align: justify;">La tassonomia di Sundevall non considerava solamente il becco e le zampe come criteri di suddivisione, ma introduceva anche il 'piumaggio' ed il 'tarso' (parte dello scheletro della&nbsp;zampa compreso tra i metatarsi e le ossa della&nbsp;zampa). Inoltre questi nuovi attributi erano vincolati in modo differente rispetto a quanto si verificava nella tassonomia di Ray.</p> <p style="text-align: justify;">La presente nota ha l'obiettivo di calcolare l'informazione semantica del dynamic frame associato alla tassonomia di Sundevall.</p> <p style="text-align: justify;">Abbiamo dunque un dynamic frame per il concetto 'bird' che presenta 4 attributi:</p> <p style="text-align: center;">Bird = {beak:[...]; foot:[...]; plumage:[...]; tarsus:[...]}</p> <p style="text-align: justify;">Per completare il dynamic frame dobbiamo indicare anche i&nbsp;valori assunti dagli attributi:</p> <p style="text-align: center;">Bird = {</p> <p style="text-align: center;">beak:[round, pointed];</p> <p style="text-align: center;">foot:[webbed, clawed];</p> <p style="text-align: center;">plumage:[dense, coarse];</p> <p style="text-align: center;">tarsus:[skinned, scutate]</p> <p style="text-align: center;">}</p> <p style="text-align: justify;">Abbiamo dunque a che fare con un dynamic frame composto da 4 attributi, ognuno dei quali pu&ograve; assumere 2 valori. Sono anche presenti dei vincoli fra i vari attributi, che sono:</p> <p style="text-align: center;">foot = clawed --&gt; tarsus = scutate + plumage = coarse</p> <p style="text-align: center;">foot = webed --&gt; tarsus = skinned + plumage = dense</p> <p style="text-align: justify;">I medesimi vincoli si hanno scambiando fra loro i valori all'interno di uno stesso vincolo; cos&igrave; ad esempio vale:</p> <p style="text-align: center;">tarsus = scutate --&gt;&nbsp;foot = clawed + plumage = coarse</p> <p style="text-align: justify;">i vincoli dunque sono simmetrici per scambio di attributo/valore.</p> <p style="text-align: justify;">Come abbiamo mostrato in alcune note precedenti un singolo attributo pu&ograve; essere descritto da un Linguaggio&nbsp;costituito da n predicati - equivalenti agli n valori assunti dall'attributo - e un'unica costante individuale (l'attributo stesso). Nel nostro caso abbiamo a che fare con 4 linguaggi&nbsp;<img title="L^2_1" src="https://latex.codecogs.com/gif.latex?L^2_1" />che portano ad avere un linguaggio&nbsp;<img title="L^8_4" src="https://latex.codecogs.com/gif.latex?L^8_4" />&nbsp;per&nbsp;l'intero dynamic frame. A questo linguaggio corrispondono&nbsp;<img title="2^{8\cdot 4}=2^{32}" src="https://latex.codecogs.com/gif.latex?2^{8\cdot&amp;space;4}=2^{32}" />&nbsp;state-description, che creano lo spazio logico del dynamic frame.</p> <p style="text-align: justify;">Sappiamo inoltre che ogni attributo possiede&nbsp;una base costituita da quei particolari state-description in cui &egrave; valido un solo predicato e che prendono il nome di base-state-description. Abbiamo anche visto come la combinazione dei base-state-description dei vari attributi porta a definire una base per l'intero dynamic frame.</p> <p style="text-align: justify;">Nel nostro caso abbiamo che la base &egrave; costituita da:</p> <p style="text-align: center;"><img title="B_{DF} = \sum_{i=1}^2 \sum_{j=1}^2 \sum_{k=1}^2 \sum_{r=1}^2 b^b_i\cup f^b_j\cup p^b_k\cup t^b_r" src="https://latex.codecogs.com/gif.latex?B_{DF}&amp;space;=&amp;space;\sum_{i=1}^2&amp;space;\sum_{j=1}^2&amp;space;\sum_{k=1}^2&amp;space;\sum_{r=1}^2&amp;space;b^b_i\cup&amp;space;f^b_j\cup&amp;space;p^b_k\cup&amp;space;t^b_r" /></p> <p style="text-align: justify;">dove&nbsp;<img title="f^b_j" src="https://latex.codecogs.com/gif.latex?f^b_j" />,&nbsp;<img title="p^b_k" src="https://latex.codecogs.com/gif.latex?p^b_k" />&nbsp;,&nbsp;<img title="t^b_r" src="https://latex.codecogs.com/gif.latex?t^b_r" />&nbsp;e&nbsp;<img title="b^b_i" src="https://latex.codecogs.com/gif.latex?b^b_i" />&nbsp;sono rispettivamente gli elementi delle basi degli attributi 'foot', 'plumage', 'tarsus' e 'beak'. Ogni elemento della base del dynamic frame &egrave; dunque rappresentato come:</p> <p style="text-align: center;"><img title="y_{ijkr} = b^b_i\cup f^b_j\cup p^b_k\cup t^b_r" src="https://latex.codecogs.com/gif.latex?y_{ijkr}&amp;space;=&amp;space;b^b_i\cup&amp;space;f^b_j\cup&amp;space;p^b_k\cup&amp;space;t^b_r" /></p> <p style="text-align: justify;">Occorre ora definire una misura di probabilit&agrave; per ogni elemento della base. Per fare questo &egrave; necessario calcolare la probabilit&agrave; congiunta:</p> <p style="text-align: center;"><img title="m_{DF}(y_{ijkr})= m_{DF}(b^b_i , f^b_j , p^b_k , t^b_r)" src="https://latex.codecogs.com/gif.latex?m_{DF}(y_{ijkr})=&amp;space;m_{DF}(b^b_i&amp;space;,&amp;space;f^b_j&amp;space;,&amp;space;p^b_k&amp;space;,&amp;space;t^b_r)" /></p> <p style="text-align: justify;">Ci vengono ora in aiuto le seguenti considerazioni:</p> <ul> <li style="text-align: justify;">La misura di probabilit&agrave; degli elementi&nbsp;<img title="b^b_i" src="https://latex.codecogs.com/gif.latex?b^b_i" />&nbsp;&egrave; indipendente dalle misure di probabilit&agrave; associate agli altri attributi; questo ci porta alla seguente semplificazione:</li> </ul> <p style="padding-left: 30px; text-align: center;"><img title="m_{DF}(b^b_i , f^b_j, p^b_k, t^b_r) = m_b(b^b_i)\cdot p(f^b_j, p^b_k, t^b_r)" src="https://latex.codecogs.com/gif.latex?m_{DF}(b^b_i&amp;space;,&amp;space;f^b_j,&amp;space;p^b_k,&amp;space;t^b_r)&amp;space;=&amp;space;m_b(b^b_i)\cdot&amp;space;p(f^b_j,&amp;space;p^b_k,&amp;space;t^b_r)" /></p> <ul> <li style="text-align: justify;">Gli elementi delle basi 'f', 'p' e 't' sono fra di loro collegati dai vincoli:</li> </ul> <p style="padding-left: 30px; text-align: center;"><img title="p(f^b_j |t^b_r) = \left \{ 1,0 \right \}" src="https://latex.codecogs.com/gif.latex?p(f^b_j&amp;space;|t^b_r)&amp;space;=&amp;space;\left&amp;space;\{&amp;space;1,0&amp;space;\right&amp;space;\}" /></p> <p style="padding-left: 30px; text-align: center;"><img title="p(p^b_j |t^b_r) = \left \{ 1,0 \right \}" src="https://latex.codecogs.com/gif.latex?p(p^b_j&amp;space;|t^b_r)&amp;space;=&amp;space;\left&amp;space;\{&amp;space;1,0&amp;space;\right&amp;space;\}" /></p> <p style="padding-left: 30px; text-align: center;"><img title="p(f^b_j | p^b_k , t^b_r) = \left \{ 1,0 \right \}" src="https://latex.codecogs.com/gif.latex?p(f^b_j&amp;space;|&amp;space;p^b_k&amp;space;,&amp;space;t^b_r)&amp;space;=&amp;space;\left&amp;space;\{&amp;space;1,0&amp;space;\right&amp;space;\}" /></p> <p style="padding-left: 30px; text-align: center;"><img title="p(p^b_k | f^b_j , t^b_r) = \left \{ 1,0 \right \}" src="https://latex.codecogs.com/gif.latex?p(p^b_k&amp;space;|&amp;space;f^b_j&amp;space;,&amp;space;t^b_r)&amp;space;=&amp;space;\left&amp;space;\{&amp;space;1,0&amp;space;\right&amp;space;\}" /></p> <p style="padding-left: 30px; text-align: justify;">dove&nbsp;i dati inseriti fra '{}' sono i possibili valori assunti dalla probabilit&agrave;. Cos&igrave; ad esempio&nbsp;valgono le relazioni seguenti:</p> <p style="padding-left: 30px; text-align: center;"><img title="p(f=webbed | t=scutate)=1" src="https://latex.codecogs.com/gif.latex?p(f=webbed&amp;space;|&amp;space;t=scutate)=1" /></p> <p style="padding-left: 30px; text-align: center;"><img title="p(f=webbed | t= skinned) = 0" src="https://latex.codecogs.com/gif.latex?p(f=webbed&amp;space;|&amp;space;t=&amp;space;skinned)&amp;space;=&amp;space;0" />&nbsp;</p> <p style="padding-left: 30px; text-align: justify;">Medesima cosa si pu&ograve; ottenere per la combinazione di altri attributi.</p> <p style="padding-left: 30px; text-align: justify;">Le ultime due condizioni sono valide in quanto l'attributo in oggetto viene sempre determinato una volta noti gli altri due. Se io ad esempio conosco il&nbsp;plumage e il tarsus, ho determinato in modo univoco anche il foot, come si evidenzia dalla relazione seguente:</p> <p style="padding-left: 30px; text-align: center;"><img title="p(f=webbed | p=dense, t=scutate) = 1" src="https://latex.codecogs.com/gif.latex?p(f=webbed&amp;space;|&amp;space;p=dense,&amp;space;t=scutate)&amp;space;=&amp;space;1" /></p> <p style="padding-left: 30px; text-align: center;"><img title="p(f=webbed | p=coarse, t=scutate) = 0" src="https://latex.codecogs.com/gif.latex?p(f=webbed&amp;space;|&amp;space;p=coarse,&amp;space;t=scutate)&amp;space;=&amp;space;0" /></p> <p style="padding-left: 30px; text-align: center;"><img title="p(f=webbed | p=dense, t=skinned) = 0" src="https://latex.codecogs.com/gif.latex?p(f=webbed&amp;space;|&amp;space;p=dense,&amp;space;t=skinned)&amp;space;=&amp;space;0" /></p> <p style="padding-left: 30px; text-align: justify;">dove le ultime due relazioni valgono in quanto i&nbsp;valori degli attributi nella probabilit&agrave; condizionata non sono tra loro compatibili.</p> <ul> <li style="text-align: justify;">Vale infine la seguente condizione per la probabilit&agrave; congiunta di 3 elementi di basi differenti:</li> </ul> <p style="padding-left: 30px; text-align: center;"><img title="p(f^b_j, p^b_k, t^b_r) = p(f^b_j | p^b_k, t^b_r)\cdot p(p^b_k | t^b_r) \cdot p(t^b_r)" src="https://latex.codecogs.com/gif.latex?p(f^b_j,&amp;space;p^b_k,&amp;space;t^b_r)&amp;space;=&amp;space;p(f^b_j&amp;space;|&amp;space;p^b_k,&amp;space;t^b_r)\cdot&amp;space;p(p^b_k&amp;space;|&amp;space;t^b_r)&amp;space;\cdot&amp;space;p(t^b_r)" /></p> <p style="padding-left: 30px; text-align: justify;">dove si &egrave; utilizzata la 'chain rule' del calcolo delle probabilit&agrave;. Imponendo ora i vincoli abbiamo:</p> <p style="padding-left: 30px; text-align: center;"><img title="p(f^b_j, p^b_k, t^b_r) = \delta_{1,0}\cdot \delta_{1,0}\cdot p(t^b_r) = p(t^b_r)" src="https://latex.codecogs.com/gif.latex?p(f^b_j,&amp;space;p^b_k,&amp;space;t^b_r)&amp;space;=&amp;space;\delta_{1,0}\cdot&amp;space;\delta_{1,0}\cdot&amp;space;p(t^b_r)&amp;space;=&amp;space;p(t^b_r)" /></p> <p style="text-align: justify;">dove <img title="\delta_{1,0}" src="https://latex.codecogs.com/gif.latex?\delta_{1,0}" />&nbsp;&egrave; una funzione che assume il valore 1 o 0 a seconda del vincolo che si sta considerando. Se ora applichiamo i risultati precedenti al caso generale abbiamo:</p> <p style="text-align: center;"><img title="m_{DF}(b^b_i, f^b_j, p^b_k, t^b_r) = m_b(b^b_i)\cdot p(f^b_j, p^b_k, t^b_r) = m_b(b^b_i)\cdot m_t(t^b_r)" src="https://latex.codecogs.com/gif.latex?m_{DF}(b^b_i,&amp;space;f^b_j,&amp;space;p^b_k,&amp;space;t^b_r)&amp;space;=&amp;space;m_b(b^b_i)\cdot&amp;space;p(f^b_j,&amp;space;p^b_k,&amp;space;t^b_r)&amp;space;=&amp;space;m_b(b^b_i)\cdot&amp;space;m_t(t^b_r)" /></p> <p style="text-align: justify;">dove si vede che la misura di probabilit&agrave; per il nostro dynamic frame &egrave; funzione solo delle misure di probabilit&agrave; sui base-state-description dell'attributo 'beak' e 'tarsus' (l'attributo 'tarsus' &egrave; stato considerato come preferenziale, ma gli stessi risultati si avrebbero anche considerando gli attributi 'plumage' e 'foot' che sono vincolati all'attributo 'tarsus').</p> <p style="text-align: justify;">Costruiamo ora gli elementi delle basi dei 4 attributi, che sono riproposti nella tabella seguente:</p> <table style="height: 61px; width: 446px;"> <tbody> <tr style="height: 13px;"> <td style="width: 36.427px; height: 13px; background-color: green; text-align: center;" colspan="4">beak</td> </tr> <tr style="height: 13px;"> <td style="width: 36.427px; height: 13px; background-color: yellow; text-align: center;">state</td> <td style="width: 140.681px; height: 13px; background-color: yellow; text-align: center;">round</td> <td style="width: 147.732px; height: 13px; background-color: yellow; text-align: center;">pointed</td> <td style="width: 98.1175px; height: 13px; background-color: yellow; text-align: center;">m</td> </tr> <tr style="height: 13.5983px; text-align: center;"> <td style="width: 36.427px; height: 13.5983px;">b1</td> <td style="width: 140.681px; height: 13.5983px;">true</td> <td style="width: 147.732px; height: 13.5983px;">false</td> <td style="width: 98.1175px; height: 13.5983px;">0.5</td> </tr> <tr style="height: 13px; text-align: center;"> <td style="width: 36.427px; height: 13px;">b2</td> <td style="width: 140.681px; height: 13px;">false</td> <td style="width: 147.732px; height: 13px;">true</td> <td style="width: 98.1175px; height: 13px;">0.5</td> </tr> <tr style="height: 13px; text-align: center;"> <td style="width: 36.427px; height: 13px; background-color: green;" colspan="4">foot</td> </tr> <tr style="height: 13px; text-align: center;"> <td style="width: 36.427px; height: 13px; background-color: yellow;">state</td> <td style="width: 140.681px; height: 13px; background-color: yellow;">webbed</td> <td style="width: 147.732px; height: 13px; background-color: yellow;">clawed</td> <td style="width: 98.1175px; height: 13px; background-color: yellow;">m</td> </tr> <tr style="height: 13px; text-align: center;"> <td style="width: 36.427px; height: 13px;">f1</td> <td style="width: 140.681px; height: 13px;">true</td> <td style="width: 147.732px; height: 13px;">false</td> <td style="width: 98.1175px; height: 13px;">0.5</td> </tr> <tr style="height: 13px; text-align: center;"> <td style="width: 36.427px; height: 13px;">f2</td> <td style="width: 140.681px; height: 13px;">false</td> <td style="width: 147.732px; height: 13px;">true</td> <td style="width: 98.1175px; height: 13px;">0.5</td> </tr> <tr style="height: 13px; text-align: center;"> <td style="width: 36.427px; height: 13px; background-color: green;" colspan="4">plumage</td> </tr> <tr style="height: 13px; text-align: center;"> <td style="width: 36.427px; height: 13px; background-color: yellow;">state</td> <td style="width: 140.681px; height: 13px; background-color: yellow;">dense</td> <td style="width: 147.732px; height: 13px; background-color: yellow;">coarse</td> <td style="width: 98.1175px; height: 13px; background-color: yellow;">m</td> </tr> <tr style="height: 13px; text-align: center;"> <td style="width: 36.427px; height: 13px;">p1</td> <td style="width: 140.681px; height: 13px;">true</td> <td style="width: 147.732px; height: 13px;">false</td> <td style="width: 98.1175px; height: 13px;">0.5</td> </tr> <tr style="height: 13px; text-align: center;"> <td style="width: 36.427px; height: 13px;">p2</td> <td style="width: 140.681px; height: 13px;">false</td> <td style="width: 147.732px; height: 13px;">true</td> <td style="width: 98.1175px; height: 13px;">0.5</td> </tr> <tr style="height: 13px; text-align: center;"> <td style="width: 36.427px; height: 13px; background-color: green;" colspan="4">tarsus</td> </tr> <tr style="height: 13px; text-align: center;"> <td style="width: 36.427px; height: 13px; background-color: yellow;">state</td> <td style="width: 140.681px; height: 13px; background-color: yellow;">skinned</td> <td style="width: 147.732px; height: 13px; background-color: yellow;">scutate</td> <td style="width: 98.1175px; height: 13px; background-color: yellow;">m</td> </tr> <tr style="height: 13px; text-align: center;"> <td style="width: 36.427px; height: 13px;">t1</td> <td style="width: 140.681px; height: 13px;">true</td> <td style="width: 147.732px; height: 13px;">false</td> <td style="width: 98.1175px; height: 13px;">0.5</td> </tr> <tr style="height: 13px; text-align: center;"> <td style="width: 36.427px; height: 13px;">t2</td> <td style="width: 140.681px; height: 13px;">false</td> <td style="width: 147.732px; height: 13px;">true</td> <td style="width: 98.1175px; height: 13px;">0.5</td> </tr> </tbody> </table> <p style="text-align: justify;">dove in verde sono indicati gli attributi, in giallo i predicati degli attributi (i valori) e la misura di probabilit&agrave; associata.</p> <p style="text-align: justify;">Se ora consideriamo la combinazione dei vari base-state-description per formare la base del dynamic frame otteniamo la tabella seguente:</p> <table style="height: 61px; width: 446px;"> <tbody> <tr style="height: 13px;"> <td style="width: 36.427px; height: 13px; background-color: yellow; text-align: center;">state</td> <td style="width: 140.681px; height: 13px; background-color: yellow; text-align: center;">combinazione di base-state-description</td> </tr> <tr style="height: 13px;"> <td style="width: 36.427px; height: 13px; background-color: yellow; text-align: center;">&nbsp;</td> <td style="width: 140.681px; height: 13px; background-color: yellow; text-align: center;">&nbsp;</td> </tr> <tr style="height: 13px; text-align: center;"> <td style="width: 36.427px; height: 13px;">y1111</td> <td style="width: 140.681px; height: 13px;">b1 + f1 + p1 + t1</td> </tr> <tr style="height: 13.3224px; text-align: center;"> <td style="width: 36.427px; height: 13.3224px;">y1112</td> <td style="width: 140.681px; height: 13.3224px;">b1 +&nbsp;f1 + p1 + t2</td> </tr> <tr style="height: 13px; text-align: center;"> <td style="width: 36.427px; height: 13px;">y1121</td> <td style="width: 140.681px; height: 13px;">b1 +&nbsp;f1 + p2 + t1</td> </tr> <tr style="height: 13px; text-align: center;"> <td style="width: 36.427px; height: 13px;">y1122</td> <td style="width: 140.681px; height: 13px;">b1 +&nbsp;f1 + p2 + t2</td> </tr> <tr style="height: 13px; text-align: center;"> <td style="width: 36.427px; height: 13px;">y1211</td> <td style="width: 140.681px; height: 13px;">b1 +&nbsp;f2 + p1 + t1</td> </tr> <tr style="height: 13px; text-align: center;"> <td style="width: 36.427px; height: 13px;">y1212</td> <td style="width: 140.681px; height: 13px;">b1 +&nbsp;f2 + p1 + t2</td> </tr> <tr style="height: 13px; text-align: center;"> <td style="width: 36.427px; height: 13px;">y1221</td> <td style="width: 140.681px; height: 13px;">b1 +&nbsp;f2 + p2 + t1</td> </tr> <tr style="height: 13px; text-align: center;"> <td style="width: 36.427px; height: 13px;">y1222</td> <td style="width: 140.681px; height: 13px;">b1 +&nbsp;f2 + p2 + t2</td> </tr> <tr style="height: 13px; text-align: center;"> <td style="width: 36.427px; height: 13px;">y2111</td> <td style="width: 140.681px; height: 13px;">b2 +&nbsp;f1 + p1 + t1</td> </tr> <tr style="height: 13px; text-align: center;"> <td style="width: 36.427px; height: 13px;">y2112</td> <td style="width: 140.681px; height: 13px;">b2 +&nbsp;f1 + p1 + t2</td> </tr> <tr style="height: 13px; text-align: center;"> <td style="width: 36.427px; height: 13px;">y2121</td> <td style="width: 140.681px; height: 13px;">b2 +&nbsp;f1 + p2 + t1</td> </tr> <tr style="height: 13px; text-align: center;"> <td style="width: 36.427px; height: 13px;">y2122</td> <td style="width: 140.681px; height: 13px;">b2 +&nbsp;f1 + p2 + t2</td> </tr> <tr style="height: 13px; text-align: center;"> <td style="width: 36.427px; height: 13px;">y2211</td> <td style="width: 140.681px; height: 13px;">b2 +&nbsp;f2 + p1 + t1</td> </tr> <tr style="height: 13px; text-align: center;"> <td style="width: 36.427px; height: 13px;">y2212</td> <td style="width: 140.681px; height: 13px;">b2 +&nbsp;f2 + p1 + t2</td> </tr> <tr style="height: 13px; text-align: center;"> <td style="width: 36.427px; height: 13px;">y2221</td> <td style="width: 140.681px; height: 13px;">b2 +&nbsp;f2 + p2 + t1</td> </tr> <tr style="height: 13px; text-align: center;"> <td style="width: 36.427px; height: 13px;">y2222</td> <td style="width: 140.681px; height: 13px;">b2 +&nbsp;f2 + p2 + t2</td> </tr> </tbody> </table> <p style="text-align: justify;">Nella tabella sono presenti 16 combinazioni. Iniziamo con il considerare i primi 8 stati e riscriviamo la tabella nel modo seguente, che &egrave; utile per calcolare la misura di probabilit&agrave;:</p> <table style="height: 58px;" width="524"> <tbody> <tr style="height: 12px;"> <td style="width: 52.8709px; background-color: yellow; text-align: center; height: 12px;">state</td> <td style="width: 52.8709px; height: 12px; background-color: green; text-align: center;" colspan="2">beak</td> <td style="width: 52.8709px; height: 12px; background-color: green; text-align: center;" colspan="2">foot</td> <td style="width: 52.8709px; height: 12px; background-color: green; text-align: center;" colspan="2">plumage</td> <td style="width: 52.8709px; height: 12px; background-color: green; text-align: center;" colspan="3">tarsus</td> </tr> <tr style="height: 13px;"> <td style="width: 52.8709px; height: 13px;">&nbsp;</td> <td style="width: 52.8709px; background-color: yellow; text-align: center; height: 13px;">round</td> <td style="width: 52.8709px; background-color: yellow; text-align: center; height: 13px;">pointed</td> <td style="width: 52.8709px; background-color: yellow; text-align: center; height: 13px;">webbed</td> <td style="width: 52.8709px; background-color: yellow; text-align: center; height: 13px;">clawed</td> <td style="width: 52.8709px; background-color: yellow; text-align: center; height: 13px;">dense</td> <td style="width: 52.8709px; background-color: yellow; text-align: center; height: 13px;">coarse</td> <td style="width: 52.8709px; background-color: yellow; text-align: center; height: 13px;">skinned</td> <td style="width: 52.8709px; background-color: yellow; text-align: center; height: 13px;">scutate</td> <td style="width: 52.8709px; background-color: yellow; text-align: center; height: 13px;">m</td> </tr> <tr style="height: 13px;"> <td style="width: 52.8709px; text-align: center; height: 13px;">y1111</td> <td style="width: 52.8709px; text-align: center; height: 13px;">true</td> <td style="width: 52.8709px; text-align: center; height: 13px;">false</td> <td style="width: 52.8709px; text-align: center; height: 13px;">true</td> <td style="width: 52.8709px; text-align: center; height: 13px;">false</td> <td style="width: 52.8709px; text-align: center; height: 13px;">true</td> <td style="width: 52.8709px; text-align: center; height: 13px;">false</td> <td style="width: 52.8709px; text-align: center; height: 13px;">true</td> <td style="width: 52.9462px; text-align: center; height: 13px;">false</td> <td style="width: 52.9462px; text-align: center; height: 13px;">0.25</td> </tr> <tr style="height: 13.7192px;"> <td style="width: 52.8709px; text-align: center; height: 13.7192px;">y1112</td> <td style="width: 52.8709px; text-align: center; height: 13.7192px;">true</td> <td style="width: 52.8709px; text-align: center; height: 13.7192px;">false</td> <td style="width: 52.8709px; text-align: center; height: 13.7192px;">true</td> <td style="width: 52.8709px; text-align: center; height: 13.7192px;">false</td> <td style="width: 52.8709px; text-align: center; height: 13.7192px;">true</td> <td style="width: 52.8709px; text-align: center; height: 13.7192px;">false</td> <td style="width: 52.8709px; text-align: center; height: 13.7192px;">false</td> <td style="width: 52.9462px; text-align: center; height: 13.7192px;">true</td> <td style="width: 52.9462px; text-align: center; height: 13.7192px;">0.0</td> </tr> <tr style="height: 13px;"> <td style="width: 52.8709px; text-align: center; height: 13px;">y1121</td> <td style="width: 52.8709px; text-align: center; height: 13px;">true</td> <td style="width: 52.8709px; text-align: center; height: 13px;">false</td> <td style="width: 52.8709px; text-align: center; height: 13px;">true</td> <td style="width: 52.8709px; text-align: center; height: 13px;">false</td> <td style="width: 52.8709px; text-align: center; height: 13px;">false</td> <td style="width: 52.8709px; text-align: center; height: 13px;">true</td> <td style="width: 52.8709px; text-align: center; height: 13px;">true</td> <td style="width: 52.9462px; text-align: center; height: 13px;">false</td> <td style="width: 52.9462px; text-align: center; height: 13px;">0.0</td> </tr> <tr style="height: 13px;"> <td style="width: 52.8709px; text-align: center; height: 13px;">y1122</td> <td style="width: 52.8709px; text-align: center; height: 13px;">true</td> <td style="width: 52.8709px; text-align: center; height: 13px;">false</td> <td style="width: 52.8709px; text-align: center; height: 13px;">true</td> <td style="width: 52.8709px; text-align: center; height: 13px;">false</td> <td style="width: 52.8709px; text-align: center; height: 13px;">false</td> <td style="width: 52.8709px; text-align: center; height: 13px;">true</td> <td style="width: 52.8709px; text-align: center; height: 13px;">false</td> <td style="width: 52.9462px; text-align: center; height: 13px;">true</td> <td style="width: 52.9462px; text-align: center; height: 13px;">0.0</td> </tr> <tr style="height: 13px;"> <td style="width: 52.8709px; text-align: center; height: 13px;">y1211</td> <td style="width: 52.8709px; text-align: center; height: 13px;">true</td> <td style="width: 52.8709px; text-align: center; height: 13px;">false</td> <td style="width: 52.8709px; text-align: center; height: 13px;">false</td> <td style="width: 52.8709px; text-align: center; height: 13px;">true</td> <td style="width: 52.8709px; text-align: center; height: 13px;">true</td> <td style="width: 52.8709px; text-align: center; height: 13px;">false</td> <td style="width: 52.8709px; text-align: center; height: 13px;">true</td> <td style="width: 52.9462px; text-align: center; height: 13px;">false</td> <td style="width: 52.9462px; text-align: center; height: 13px;">0.0</td> </tr> <tr style="height: 13px;"> <td style="width: 52.8709px; text-align: center; height: 13px;">y1212</td> <td style="width: 52.8709px; text-align: center; height: 13px;">true</td> <td style="width: 52.8709px; text-align: center; height: 13px;">false</td> <td style="width: 52.8709px; text-align: center; height: 13px;">false</td> <td style="width: 52.8709px; text-align: center; height: 13px;">true</td> <td style="width: 52.8709px; text-align: center; height: 13px;">true</td> <td style="width: 52.8709px; text-align: center; height: 13px;">false</td> <td style="width: 52.8709px; text-align: center; height: 13px;">false</td> <td style="width: 52.9462px; text-align: center; height: 13px;">true</td> <td style="width: 52.9462px; text-align: center; height: 13px;">0.0</td> </tr> <tr style="height: 13px;"> <td style="width: 52.8709px; text-align: center; height: 13px;">y1221</td> <td style="width: 52.8709px; text-align: center; height: 13px;">true</td> <td style="width: 52.8709px; text-align: center; height: 13px;">false</td> <td style="width: 52.8709px; text-align: center; height: 13px;">false</td> <td style="width: 52.8709px; text-align: center; height: 13px;">true</td> <td style="width: 52.8709px; text-align: center; height: 13px;">false</td> <td style="width: 52.8709px; text-align: center; height: 13px;">true</td> <td style="width: 52.8709px; text-align: center; height: 13px;">true</td> <td style="width: 52.9462px; text-align: center; height: 13px;">false</td> <td style="width: 52.9462px; text-align: center; height: 13px;">0.0</td> </tr> <tr style="height: 13px;"> <td style="width: 52.8709px; text-align: center; height: 13px;">y1222</td> <td style="width: 52.8709px; text-align: center; height: 13px;">true</td> <td style="width: 52.8709px; text-align: center; height: 13px;">false</td> <td style="width: 52.8709px; text-align: center; height: 13px;">false</td> <td style="width: 52.8709px; text-align: center; height: 13px;">true</td> <td style="width: 52.8709px; text-align: center; height: 13px;">false</td> <td style="width: 52.8709px; text-align: center; height: 13px;">true</td> <td style="width: 52.8709px; text-align: center; height: 13px;">false</td> <td style="width: 52.9462px; text-align: center; height: 13px;">true</td> <td style="width: 52.9462px; text-align: center; height: 13px;">0.25</td> </tr> </tbody> </table> <p style="text-align: justify;">Nell'ultima colonna &egrave; presente la misura di probabilit&agrave; associata allo stato. Si noti che molte combinazioni non sono ammesse; ad esempio la misura di probabilit&agrave;&nbsp;per&nbsp;lo stato y1121 &egrave;:</p> <p style="text-align: justify;"><img title="p(b=round, f = webbed, p = coarse, t = skinned) = p(b=round) * p(f=round | p=coarse, t=skinned) * p(p=coarse | t = skinned) * p(t=skinned)" src="https://latex.codecogs.com/gif.latex?p(b=round,&amp;space;f&amp;space;=&amp;space;webbed,&amp;space;p&amp;space;=&amp;space;coarse,&amp;space;t&amp;space;=&amp;space;skinned)&amp;space;=&amp;space;p(b=round)&amp;space;*&amp;space;p(f=round&amp;space;|&amp;space;p=coarse,&amp;space;t=skinned)&amp;space;*&amp;space;p(p=coarse&amp;space;|&amp;space;t&amp;space;=&amp;space;skinned)&amp;space;*&amp;space;p(t=skinned)" /></p> <p style="text-align: justify;">si noti che p(p=coarse | t = skinned) = 0.</p> <p style="text-align: justify;">Rimangono ora da valutare gli altri 8 stati, che portano alla tabella:</p> <table style="height: 58px;" width="524"> <tbody> <tr style="height: 12px;"> <td style="width: 52.8709px; background-color: yellow; text-align: center; height: 12px;">state</td> <td style="width: 52.8709px; height: 12px; background-color: green; text-align: center;" colspan="2">beak</td> <td style="width: 52.8709px; height: 12px; background-color: green; text-align: center;" colspan="2">foot</td> <td style="width: 52.8709px; height: 12px; background-color: green; text-align: center;" colspan="2">plumage</td> <td style="width: 52.8709px; height: 12px; background-color: green; text-align: center;" colspan="3">tarsus</td> </tr> <tr style="height: 13px;"> <td style="width: 52.8709px; height: 13px;">&nbsp;</td> <td style="width: 52.8709px; background-color: yellow; text-align: center; height: 13px;">round</td> <td style="width: 52.8709px; background-color: yellow; text-align: center; height: 13px;">pointed</td> <td style="width: 52.8709px; background-color: yellow; text-align: center; height: 13px;">webbed</td> <td style="width: 52.8709px; background-color: yellow; text-align: center; height: 13px;">clawed</td> <td style="width: 52.8709px; background-color: yellow; text-align: center; height: 13px;">dense</td> <td style="width: 52.8709px; background-color: yellow; text-align: center; height: 13px;">coarse</td> <td style="width: 52.8709px; background-color: yellow; text-align: center; height: 13px;">skinned</td> <td style="width: 52.8709px; background-color: yellow; text-align: center; height: 13px;">scutate</td> <td style="width: 52.8709px; background-color: yellow; text-align: center; height: 13px;">m</td> </tr> <tr style="height: 13.5938px;"> <td style="width: 52.8709px; text-align: center; height: 13.5938px;">y2111</td> <td style="width: 52.8709px; text-align: center; height: 13.5938px;">false</td> <td style="width: 52.8709px; text-align: center; height: 13.5938px;">true</td> <td style="width: 52.8709px; text-align: center; height: 13.5938px;">true</td> <td style="width: 52.8709px; text-align: center; height: 13.5938px;">false</td> <td style="width: 52.8709px; text-align: center; height: 13.5938px;">true</td> <td style="width: 52.8709px; text-align: center; height: 13.5938px;">false</td> <td style="width: 52.8709px; text-align: center; height: 13.5938px;">true</td> <td style="width: 52.9462px; text-align: center; height: 13.5938px;">false</td> <td style="width: 52.9462px; text-align: center; height: 13.5938px;">0.25</td> </tr> <tr style="height: 13px;"> <td style="width: 52.8709px; text-align: center; height: 13px;">y2112</td> <td style="width: 52.8709px; text-align: center; height: 13px;">false</td> <td style="width: 52.8709px; text-align: center; height: 13px;">true</td> <td style="width: 52.8709px; text-align: center; height: 13px;">true</td> <td style="width: 52.8709px; text-align: center; height: 13px;">false</td> <td style="width: 52.8709px; text-align: center; height: 13px;">true</td> <td style="width: 52.8709px; text-align: center; height: 13px;">false</td> <td style="width: 52.8709px; text-align: center; height: 13px;">false</td> <td style="width: 52.9462px; text-align: center; height: 13px;">true</td> <td style="width: 52.9462px; text-align: center; height: 13px;">0.0</td> </tr> <tr style="height: 13px;"> <td style="width: 52.8709px; text-align: center; height: 13px;">y2121</td> <td style="width: 52.8709px; text-align: center; height: 13px;">false</td> <td style="width: 52.8709px; text-align: center; height: 13px;">true</td> <td style="width: 52.8709px; text-align: center; height: 13px;">true</td> <td style="width: 52.8709px; text-align: center; height: 13px;">false</td> <td style="width: 52.8709px; text-align: center; height: 13px;">false</td> <td style="width: 52.8709px; text-align: center; height: 13px;">true</td> <td style="width: 52.8709px; text-align: center; height: 13px;">true</td> <td style="width: 52.9462px; text-align: center; height: 13px;">false</td> <td style="width: 52.9462px; text-align: center; height: 13px;">0.0</td> </tr> <tr style="height: 13px;"> <td style="width: 52.8709px; text-align: center; height: 13px;">y2122</td> <td style="width: 52.8709px; text-align: center; height: 13px;">false</td> <td style="width: 52.8709px; text-align: center; height: 13px;">true</td> <td style="width: 52.8709px; text-align: center; height: 13px;">true</td> <td style="width: 52.8709px; text-align: center; height: 13px;">false</td> <td style="width: 52.8709px; text-align: center; height: 13px;">false</td> <td style="width: 52.8709px; text-align: center; height: 13px;">true</td> <td style="width: 52.8709px; text-align: center; height: 13px;">false</td> <td style="width: 52.9462px; text-align: center; height: 13px;">true</td> <td style="width: 52.9462px; text-align: center; height: 13px;">0.0</td> </tr> <tr style="height: 13px;"> <td style="width: 52.8709px; text-align: center; height: 13px;">y2211</td> <td style="width: 52.8709px; text-align: center; height: 13px;">false</td> <td style="width: 52.8709px; text-align: center; height: 13px;">true</td> <td style="width: 52.8709px; text-align: center; height: 13px;">false</td> <td style="width: 52.8709px; text-align: center; height: 13px;">true</td> <td style="width: 52.8709px; text-align: center; height: 13px;">true</td> <td style="width: 52.8709px; text-align: center; height: 13px;">false</td> <td style="width: 52.8709px; text-align: center; height: 13px;">true</td> <td style="width: 52.9462px; text-align: center; height: 13px;">false</td> <td style="width: 52.9462px; text-align: center; height: 13px;">0.0</td> </tr> <tr style="height: 13px;"> <td style="width: 52.8709px; text-align: center; height: 13px;">y2212</td> <td style="width: 52.8709px; text-align: center; height: 13px;">false</td> <td style="width: 52.8709px; text-align: center; height: 13px;">true</td> <td style="width: 52.8709px; text-align: center; height: 13px;">false</td> <td style="width: 52.8709px; text-align: center; height: 13px;">true</td> <td style="width: 52.8709px; text-align: center; height: 13px;">true</td> <td style="width: 52.8709px; text-align: center; height: 13px;">false</td> <td style="width: 52.8709px; text-align: center; height: 13px;">false</td> <td style="width: 52.9462px; text-align: center; height: 13px;">true</td> <td style="width: 52.9462px; text-align: center; height: 13px;">0.0</td> </tr> <tr style="height: 13px;"> <td style="width: 52.8709px; text-align: center; height: 13px;">y2221</td> <td style="width: 52.8709px; text-align: center; height: 13px;">false</td> <td style="width: 52.8709px; text-align: center; height: 13px;">true</td> <td style="width: 52.8709px; text-align: center; height: 13px;">false</td> <td style="width: 52.8709px; text-align: center; height: 13px;">true</td> <td style="width: 52.8709px; text-align: center; height: 13px;">false</td> <td style="width: 52.8709px; text-align: center; height: 13px;">true</td> <td style="width: 52.8709px; text-align: center; height: 13px;">true</td> <td style="width: 52.9462px; text-align: center; height: 13px;">false</td> <td style="width: 52.9462px; text-align: center; height: 13px;">0.0</td> </tr> <tr style="height: 13px;"> <td style="width: 52.8709px; text-align: center; height: 13px;">y2222</td> <td style="width: 52.8709px; text-align: center; height: 13px;">false</td> <td style="width: 52.8709px; text-align: center; height: 13px;">true</td> <td style="width: 52.8709px; text-align: center; height: 13px;">false</td> <td style="width: 52.8709px; text-align: center; height: 13px;">true</td> <td style="width: 52.8709px; text-align: center; height: 13px;">false</td> <td style="width: 52.8709px; text-align: center; height: 13px;">true</td> <td style="width: 52.8709px; text-align: center; height: 13px;">false</td> <td style="width: 52.9462px; text-align: center; height: 13px;">true</td> <td style="width: 52.9462px; text-align: center; height: 13px;">0.25</td> </tr> </tbody> </table> <p style="text-align: justify;">Si noti che la somma dei valori assunti dalla misura di probabilit&agrave; delle 2 tabelle &egrave; pari a 1, come ci aspetteremmo. Abbiamo dunque a che fare con solo 4 base-state-description che riassumiamo nella tabella seguente:</p> <table style="height: 110px;" width="521"> <tbody> <tr style="height: 16px;"> <td style="width: 81.3159px; height: 16px; background-color: yellow; text-align: center;">state</td> <td style="width: 81.3159px; height: 16px; background-color: yellow; text-align: center;">beak</td> <td style="width: 81.3159px; height: 16px; background-color: yellow; text-align: center;">foot</td> <td style="width: 81.3159px; height: 16px; background-color: yellow; text-align: center;">plumage</td> <td style="width: 81.3159px; height: 16px; background-color: yellow; text-align: center;">tarsus</td> <td style="width: 81.363px; height: 16px; background-color: yellow; text-align: center;">m</td> <td style="width: 81.363px; height: 16px; background-color: yellow; text-align: center;">inf</td> </tr> <tr style="height: 16.0231px;"> <td style="width: 81.3159px; height: 16.0231px; text-align: center;">y1111</td> <td style="width: 81.3159px; height: 16.0231px; text-align: center;">round</td> <td style="width: 81.3159px; height: 16.0231px; text-align: center;">webbed</td> <td style="width: 81.3159px; height: 16.0231px; text-align: center;">dense</td> <td style="width: 81.3159px; height: 16.0231px; text-align: center;">skinned</td> <td style="width: 81.363px; height: 16.0231px; text-align: center;">0.25</td> <td style="width: 81.363px; height: 16.0231px; text-align: center;">2</td> </tr> <tr style="height: 16px;"> <td style="width: 81.3159px; height: 16px; text-align: center;">y1222</td> <td style="width: 81.3159px; height: 16px; text-align: center;">round</td> <td style="width: 81.3159px; height: 16px; text-align: center;">clawed</td> <td style="width: 81.3159px; height: 16px; text-align: center;">coarse</td> <td style="width: 81.3159px; height: 16px; text-align: center;">skutate</td> <td style="width: 81.363px; height: 16px; text-align: center;">0.25</td> <td style="width: 81.363px; height: 16px; text-align: center;">2</td> </tr> <tr style="height: 16px;"> <td style="width: 81.3159px; height: 16px; text-align: center;">y2111</td> <td style="width: 81.3159px; height: 16px; text-align: center;">pointed</td> <td style="width: 81.3159px; height: 16px; text-align: center;">webbed</td> <td style="width: 81.3159px; height: 16px; text-align: center;">dense</td> <td style="width: 81.3159px; height: 16px; text-align: center;">skinned</td> <td style="width: 81.363px; height: 16px; text-align: center;">0.25</td> <td style="width: 81.363px; height: 16px; text-align: center;">2</td> </tr> <tr style="height: 16px;"> <td style="width: 81.3159px; height: 16px; text-align: center;">y2222</td> <td style="width: 81.3159px; height: 16px; text-align: center;">pointed</td> <td style="width: 81.3159px; height: 16px; text-align: center;">clawed</td> <td style="width: 81.3159px; height: 16px; text-align: center;">coarse</td> <td style="width: 81.3159px; height: 16px; text-align: center;">scutate</td> <td style="width: 81.363px; height: 16px; text-align: center;">0.25</td> <td style="width: 81.363px; height: 16px; text-align: center;">2</td> </tr> </tbody> </table> <p>Nell'ultima colonna della tabella &egrave; stato anche inserita la quantit&agrave; di informazione associata allo stato, secondo la formula:</p> <p style="text-align: center;"><img title="inf(y^b_{ijkr}) = - Log(m(y^b_{ijkr}))" src="https://latex.codecogs.com/gif.latex?inf(y^b_{ijkr})&amp;space;=&amp;space;-&amp;space;Log(m(y^b_{ijkr}))" /></p> <p style="text-align: justify;">dove si vede che per ogni stato sono necessari 2 bit di informazione (beak + (tarsus | plumage | foot)).</p> <p style="text-align: justify;">Siamo infine in grado di calcolare la quantit&agrave; di informazione associata al dynamic frame della tassonomia di Sundevall utilizzando la formula:</p> <p style="text-align: center;"><img title="est(inf, bird)= \sum_{i=1}^2 \sum_{j=1}^2 \sum_{k=1}^2 \sum_{r=1}^2 m_{DF}(y^b_{ijkr})\cdot inf(y^b_{ijkr})" src="https://latex.codecogs.com/gif.latex?est(inf,&amp;space;bird)=&amp;space;\sum_{i=1}^2&amp;space;\sum_{j=1}^2&amp;space;\sum_{k=1}^2&amp;space;\sum_{r=1}^2&amp;space;m_{DF}(y^b_{ijkr})\cdot&amp;space;inf(y^b_{ijkr})" /></p> <p style="text-align: justify;">che porta al risultato:</p> <p style="text-align: center;"><img title="est(inf, bird)= 4.0 \cdot 0.25\cdot 2.0 = 2" src="https://latex.codecogs.com/gif.latex?est(inf,&amp;space;bird)=&amp;space;4.0&amp;space;\cdot&amp;space;0.25\cdot&amp;space;2.0&amp;space;=&amp;space;2" /></p> <p style="text-align: justify;">&nbsp;</p>