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Gower’s General Similarity Coefficient is one of the most popular measures of distance for mixed data sets
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\[ S_{ij} = \frac{ \sum_{k}^{n} W_{ijk}S_{ijk} }{ \sum_{k}^{n} W_{ijk} } \]
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\[ S_{ijk} = 1 - |x_{ik} - x_{jk}| / r_{k} \]
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\(r_{k}\) is the range of values for kth variable
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Thus for identical values \(S_{ijk}\) is 1
And 0 for \(x_{max}-x_{min}\)
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| Cases | ||||
|---|---|---|---|---|
| case i | \(+\) | \(+\) | \(-\) | \(-\) |
| case j | \(+\) | \(-\) | \(+\) | \(-\) |
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| Values | ||||
|---|---|---|---|---|
| \(S_{ijk}\) | 1 | 0 | 0 | 0 |
| \(W_{ijk}\) | 1 | 1 | 1 | 0 |
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If \(k\) is a categorical variable with \(n\) levels
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| ID# | A | B |
|---|---|---|
| 1 | 2 | 1 |
| 2 | 1 | 2 |
| 3 | 3 | 2 |
| 4 | 1 | 1 |
| 5 | 2 | 1 |
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\(k\) is converted to \(n\) binary variables
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| ID# | \(A_{1}\) | \(A_{2}\) | \(A_{3}\) | \(B_{1}\) | \(B_{2}\) |
|---|---|---|---|---|---|
| 1 | \(-\) | \(+\) | \(-\) | \(+\) | \(-\) |
| 2 | \(+\) | \(-\) | \(-\) | \(-\) | \(+\) |
| 3 | \(-\) | \(-\) | \(+\) | \(-\) | \(+\) |
| 4 | \(+\) | \(-\) | \(-\) | \(+\) | \(-\) |
| 5 | \(-\) | \(+\) | \(-\) | \(+\) | \(-\) |
## Dissimilarities :
## Burrito Chalupa Chilaquiles Chimichangas Enchilada Entomatadas Gorditas Gringas Huaraches Memela Molletes Molotes Nachos Pambazos Panucho Papadzules Pastel azteca Quesadillas Salbutes Sincronizadas Tacos Tamales Taquitos Tlacoyos
## Chalupa 0.4000000
## Chilaquiles 0.8000000 0.8500000
## Chimichangas 0.1666667 0.5200000 0.6666667
## Enchilada 0.4000000 0.4400000 0.4333333 0.3428571
## Entomatadas 0.4000000 0.4400000 0.4333333 0.3428571 0.1428571
## Gorditas 0.2500000 0.1500000 1.0000000 0.4000000 0.4800000 0.4800000
## Gringas 0.5200000 0.5000000 0.6800000 0.6571429 0.5333333 0.5333333 0.6500000
## Huaraches 1.0000000 0.6000000 0.2500000 0.8000000 0.7200000 0.7200000 0.7500000 0.8500000
## Memela 1.0000000 0.6000000 0.5000000 0.8000000 0.7200000 0.7200000 0.7500000 0.8500000 0.2500000
## Molletes 1.0000000 0.8500000 0.5000000 0.8000000 0.7200000 0.7200000 1.0000000 0.8500000 0.5000000 0.5000000
## Molotes 0.2500000 0.1500000 1.0000000 0.3333333 0.4800000 0.4800000 0.0000000 0.7200000 0.7500000 0.7500000 1.0000000
## Nachos 1.0000000 0.8500000 0.5000000 0.8000000 0.7200000 0.7200000 1.0000000 0.8500000 0.5000000 0.5000000 0.5000000 1.0000000
## Pambazos 0.4500000 0.3000000 0.8400000 0.5428571 0.5666667 0.5666667 0.4500000 0.6400000 0.8400000 0.8400000 0.6400000 0.3600000 0.8400000
## Panucho 0.5000000 0.7200000 0.5000000 0.4285714 0.4857143 0.4857143 0.6000000 0.4333333 0.6000000 0.8000000 0.8000000 0.6000000 0.8000000 0.8000000
## Papadzules 0.5666667 0.6400000 0.4333333 0.4857143 0.1428571 0.2857143 0.6800000 0.5333333 0.7200000 0.7200000 0.7200000 0.6800000 0.7200000 0.7333333 0.4857143
## Pastel azteca 0.5600000 0.5500000 0.4400000 0.6857143 0.4000000 0.4000000 0.7000000 0.5333333 0.8000000 0.8000000 0.8000000 0.7600000 0.8000000 0.6000000 0.6333333 0.4000000
## Quesadillas 0.3200000 0.2500000 0.6800000 0.5142857 0.3666667 0.3666667 0.4000000 0.1666667 0.8500000 0.8500000 0.8500000 0.5200000 0.8500000 0.4400000 0.4333333 0.5333333 0.5333333
## Salbutes 0.8000000 0.8500000 0.2000000 0.5714286 0.6000000 0.6000000 1.0000000 0.5666667 0.2500000 0.5000000 0.5000000 0.8000000 0.5000000 0.7000000 0.3333333 0.6000000 0.7000000 0.5666667
## Sincronizadas 0.4666667 0.6400000 0.5333333 0.4750000 0.4857143 0.4857143 0.7600000 0.4571429 0.6400000 0.6400000 0.6400000 0.8000000 0.6400000 0.7142857 0.5428571 0.4857143 0.5714286 0.4571429 0.6000000
## Tacos 0.3200000 0.2500000 0.6800000 0.4333333 0.3666667 0.3666667 0.4000000 0.2000000 0.8500000 0.8500000 0.8500000 0.4000000 0.8500000 0.3000000 0.4333333 0.5333333 0.4400000 0.0000000 0.4800000 0.5333333
## Tamales 0.2500000 0.1500000 1.0000000 0.5000000 0.4800000 0.4800000 0.0000000 0.7200000 0.7500000 0.7500000 1.0000000 0.2000000 1.0000000 0.5600000 0.6000000 0.6800000 0.7600000 0.5200000 1.0000000 0.8000000 0.4000000
## Taquitos 0.4000000 0.4400000 0.4333333 0.1750000 0.1428571 0.1428571 0.4800000 0.6000000 0.7200000 0.7200000 0.7200000 0.4000000 0.7200000 0.4857143 0.4857143 0.2857143 0.4857143 0.4571429 0.5142857 0.5500000 0.3666667 0.5666667
## Tlacoyos 0.5000000 0.4000000 1.0000000 0.6000000 0.6800000 0.6800000 0.2500000 0.6500000 0.7500000 0.7500000 1.0000000 0.2500000 1.0000000 0.7000000 0.6000000 0.6800000 0.7000000 0.6500000 1.0000000 0.7600000 0.6500000 0.2500000 0.6800000
## Tlayudas 0.8000000 0.8500000 0.2000000 0.6666667 0.6000000 0.6000000 1.0000000 0.4800000 0.2500000 0.5000000 0.5000000 1.0000000 0.5000000 0.8400000 0.3333333 0.6000000 0.6400000 0.4800000 0.0000000 0.5333333 0.4800000 1.0000000 0.6000000 1.0000000
##
## Metric : mixed ; Types = N, N, I, A, N, A, N, N
## Number of objects : 25