Comparison of Analogy-Based Methods for Predicting Preferences

Abstract

Given a set of preferences between items taken by pairs and described in terms of nominal or numerical attribute values, the problem considered is to predict the preference between the items of a new pair. The paper proposes and compares two approaches based on analogical proportions, which are statements of the form “a is to b as c is to d”. The first one uses triples of pairs of items for which preferences are known and which make analogical proportions, altogether with the new pair. These proportions express attribute by attribute that the change of values between the items of the first two pairs is the same as between the last two pairs. This provides a basis for predicting the preference associated with the fourth pair, also making sure that no contradictory trade-offs are created. Moreover, we also consider the option that one of the pairs in the triples is taken as a k-nearest neighbor of the new pair. The second approach exploits pairs of compared items one by one: for predicting the preference between two items, one looks for another pair of items for which the preference is known such that, attribute by attribute, the change between the elements of the first pair is the same as between the elements of the second pair. As discussed in the paper, the two approaches agree with the postulates underlying weighted averages and more general multiple criteria aggregation models. The paper proposes new algorithms for implementing these methods. The reported experiments, both on real data sets and on generated datasets suggest the effectiveness of the approaches. We also compare with predictions given by weighted sums compatible with the data, and obtained by linear programming.

Appendices

A Tversky’s Additive Difference Model

Tversky’s additive difference model functions used in the experiments are given below. Let $d^1, d^2$ be a pair of alternative that have to be compared. We denote by $\eta _i$ the difference between $d^1$ and $d^2$ on the criterion i, i.e. $\eta _i = d^1_i - d^2_i$. For the TV dataset in which 3 features are involved, we used the following piecewise linear functions:

$$\begin{aligned} \varPhi _1(\eta _1)&= {\left\{ \begin{array}{ll} {{\,\mathrm{sgn}\,}}(\eta _1) \ 0.453 \cdot 0.143 \cdot \eta _1 &{} \text {if } |\eta _1| \in [0, 0.25],\\ {{\,\mathrm{sgn}\,}}(\eta _1) \ 0.453 \cdot [-0.168 + 0.815 \cdot \eta _1] &{} \text {if } |\eta _1| \in [0.25, 0.5],\\ {{\,\mathrm{sgn}\,}}(\eta _1) \ 0.453 \cdot [0.230 + 0.018 \cdot \eta _1] &{} \text {if } |\eta _1| \in [0.5, 0.75],\\ {{\,\mathrm{sgn}\,}}(\eta _1) \ 0.453 \cdot [-2.024 + 3.024 \cdot \eta _1] &{} \text {if } |\eta _1| \in [0.75, 1], \end{array}\right. }\\ \varPhi _2(\eta _2)&= {\left\{ \begin{array}{ll} {{\,\mathrm{sgn}\,}}(\eta _2) \ 0.053 \cdot 2.648 \cdot \eta _2 &{} \text {if } |\eta _2| \in [0, 0.25],\\ {{\,\mathrm{sgn}\,}}(\eta _2) \ 0.053 \cdot [0.371 + 1.163 \cdot \eta _2] &{} \text {if } |\eta _2| \in [0.25, 0.5],\\ {{\,\mathrm{sgn}\,}}(\eta _2) \ 0.053 \cdot [0.926 + 0.054 \cdot \eta _2] &{} \text {if } |\eta _2| \in [0.5, 0.75],\\ {{\,\mathrm{sgn}\,}}(\eta _2) \ 0.053 \cdot [0.866 + 0.134 \cdot \eta _2] &{} \text {if } |\eta _2| \in [0.75, 1], \end{array}\right. }\\ \varPhi _3(\eta _3)&= {\left\{ \begin{array}{ll} {{\,\mathrm{sgn}\,}}(\eta _3) \ 0.494 \cdot 0.289 \cdot \eta _3 &{} \text {if } |\eta _3| \in [0, 0.25],\\ {{\,\mathrm{sgn}\,}}(\eta _3) \ 0.494 \cdot [-0.197 + 1.076 \cdot \eta _3] &{} \text {if } |\eta _3| \in [0.25, 0.5],\\ {{\,\mathrm{sgn}\,}}(\eta _3) \ 0.494 \cdot [0.150 + 0.383 \cdot \eta _3] &{} \text {if } |\eta _3| \in [0.5, 0.75],\\ {{\,\mathrm{sgn}\,}}(\eta _3) \ 0.494 \cdot [-1.252 + 2.252 \cdot \eta _3] &{} \text {if } |\eta _3| \in [0.75, 1]. \end{array}\right. } \end{aligned}$$

For the TV dataset in which 5 features are involved, we used the following piecewise linear functions:

$$\begin{aligned} \varPhi _1(\eta _1)&= {\left\{ \begin{array}{ll} {{\,\mathrm{sgn}\,}}(\eta _1) \cdot 0.294 \cdot 2.510 \cdot \eta _1 &{} \text {if } |\eta _1| \in [0, 0.25],\\ {{\,\mathrm{sgn}\,}}(\eta _1) \cdot 0.294 \cdot [0.562 + 0.263 \cdot \eta _1] &{} \text {if } |\eta _1| \in [0.25, 0.5],\\ {{\,\mathrm{sgn}\,}}(\eta _1) \cdot 0.294 \cdot [0.645 + 0.096 \cdot \eta _1] &{} \text {if } |\eta _1| \in [0.5, 0.75],\\ {{\,\mathrm{sgn}\,}}(\eta _1) \cdot 0.294 \cdot [-0.130 + 1.130 \cdot \eta _1] &{} \text {if } |\eta _1| \in [0.75, 1], \end{array}\right. }\\ \varPhi _2(\eta _2)&= {\left\{ \begin{array}{ll} {{\,\mathrm{sgn}\,}}(\eta _2) \cdot 0.151 \cdot 0.125 \cdot \eta _2 &{} \text {if } |\eta _2| \in [0, 0.25],\\ {{\,\mathrm{sgn}\,}}(\eta _2) \cdot 0.151 \cdot [0.025 + 0.023 \cdot \eta _2] &{} \text {if } |\eta _2| \in [0.25, 0.5],\\ {{\,\mathrm{sgn}\,}}(\eta _2) \cdot 0.151 \cdot [-0.545 + 1.164 \cdot \eta _2] &{} \text {if } |\eta _2| \in [0.5, 0.75],\\ {{\,\mathrm{sgn}\,}}(\eta _2) \cdot 0.151 \cdot [-1.689 + 2.689 \cdot \eta _2] &{} \text {if } |\eta _2| \in [0.75, 1], \end{array}\right. }\\ \varPhi _3(\eta _3)&= {\left\{ \begin{array}{ll} {{\,\mathrm{sgn}\,}}(\eta _3) \cdot 0.039 \cdot 2.388 \cdot \eta _3 &{} \text {if } |\eta _3| \in [0, 0.25],\\ {{\,\mathrm{sgn}\,}}(\eta _3) \cdot 0.039 \cdot [0.582 + 0.057 \cdot \eta _3] &{} \text {if } |\eta _3| \in [0.25, 0.5],\\ {{\,\mathrm{sgn}\,}}(\eta _3) \cdot 0.039 \cdot [-0.046 + 1.314 \cdot \eta _3] &{} \text {if } |\eta _3| \in [0.5, 0.75],\\ {{\,\mathrm{sgn}\,}}(\eta _3) \cdot 0.039 \cdot [ 0.759 + 0.241 \cdot \eta _3] &{} \text {if } |\eta _3| \in [0.75, 1], \end{array}\right. }\\ \varPhi _1(\eta _4)&= {\left\{ \begin{array}{ll} {{\,\mathrm{sgn}\,}}(\eta _4) \cdot 0.425 \cdot 0.014 \cdot \eta _4 &{} \text {if } |\eta _4| \in [0, 0.25],\\ {{\,\mathrm{sgn}\,}}(\eta _4) \cdot 0.425 \cdot [-0.110 + 0.455 \cdot \eta _4] &{} \text {if } |\eta _4| \in [0.25, 0.5],\\ {{\,\mathrm{sgn}\,}}(\eta _4) \cdot 0.425 \cdot [-0.341 + 0.917 \cdot \eta _4] &{} \text {if } |\eta _4| \in [0.5, 0.75],\\ {{\,\mathrm{sgn}\,}}(\eta _4) \cdot 0.425 \cdot [-1.613 + 2.613 \cdot \eta _4] &{} \text {if } |\eta _4| \in [0.75, 1]. \end{array}\right. }\\ \varPhi _1(\eta _5)&= {\left\{ \begin{array}{ll} {{\,\mathrm{sgn}\,}}(\eta _5) \cdot 0.091 \cdot 3.307 \cdot \eta _5 &{} \text {if } |\eta _5| \in [0, 0.25],\\ {{\,\mathrm{sgn}\,}}(\eta _5) \cdot 0.091 \cdot [0.697 + 0.519 \cdot \eta _5] &{} \text {if } |\eta _5| \in [0.25, 0.5],\\ {{\,\mathrm{sgn}\,}}(\eta _5) \cdot 0.091 \cdot [0.880 + 0.153 \cdot \eta _5] &{} \text {if } |\eta _5| \in [0.5, 0.75],\\ {{\,\mathrm{sgn}\,}}(\eta _5) \cdot 0.091 \cdot [0.979 + 0.021 \cdot \eta _5] &{} \text {if } |\eta _5| \in [0.75, 1]. \end{array}\right. } \end{aligned}$$

B Linear Program Used for Computing a Weighted Sum

We compared the performances of the algorithms presented in this paper to the results obtained with a linear program inferring the parameters of a weighted sum that fits as well as possible with the learning set. The linear program is given below:

$$\begin{aligned} \begin{array}{rlr} \min \sum _{a \in E} &{} \delta _a\\ \sum _{i = 1}^n w_i \cdot (a^1_i - a^2_i) + \delta _{a} &{} \ge 0 &{} \forall a \in E: a^1 \succcurlyeq a^2\\ \sum _{i = 1}^n w_i \cdot (a^1_i - a^2_i) - \delta _{a} &{} \le \epsilon &{} \forall a \in E: a^1 \prec a^2\\ w_i &{} \in [0,1] &{} i = {1, ..., n}\\ \delta _{a} &{} \in [0, \infty [ \end{array} \end{aligned}$$

with:

• n: number of features,

• E: learning set composed of pairs ($a_1, a_2$) evaluated on n features and a preference relation for each pair ($a_1 \succ a_2$ or $a_1 \prec a_2$),

• $w_i$: weight associated to feature i,

• $\epsilon$: a small positive value.