Case-based decision model matches ideal point model:

Abstract

This paper studies the relationship between a case-based decision theory (CBDT) and an ideal point model (IPM). We show that a case-based decision model (CBDM) can be transformed into an IPM under some assumptions. This transformation can allow us to visualize the relationship among data and simplify the calculations of distance between one current datum and the ideal point, rather than the distances between data. Our results will assist researchers with their product design analysis and positioning of goods through CBDT, by revealing past dependences or providing a reference point. Furthermore, to check whether the similarity function, presented in the theoretical part, is valid for empirical analysis, we use data on the viewing behavior of audiences of TV dramas in Japan and compare the estimation results under the CBDM that corresponds to a standard decision model with similarities and other various similarity functions and without a similarity function. Our empirical analysis shows that the CBDM with a similarity function, presented in this study, best fits the data.

Appendices

Appendix A

1.1 A.1 Proof of Proposition 1

If u _i = 1, we have the following:

$$\begin{array}{@{}rcl@{}} \sum\limits_{i=1}^{n} s(x,x_{i}) &\,=\,&f[(x\,-\,x_{1} )^{2}] \,+\, f[(x-x_{2} )^{2}]\,+\,{\cdots} \,+\,f[(x\,-\,x_{n} )^{2}] \,+\,n\beta \,=\, f\left[ \sum\limits_{i=1}^{n} (x\,-\,x_{i} )^{2} \right] \,+\,n\beta \\ &\,=\,&f\left[ nx^{2} \,-\,2x \sum\limits_{i=1}^{n} x_{i} \,+\,\sum\limits_{i=1}^{n} {x_{i}^{2}} \right] \,+\,n\beta \,=\,n f\left[ x^{2} \,-\,\frac{2{\sum}_{i=1}^{n} x_{i} }{n} x\,+\, \frac{{\sum}_{i=1}^{n} {x_{i}^{2}} }{n} \right] \,+\,n\beta \\ &\,=\,&nf\left[ \left( x\,-\, \frac{{\sum}_{i=1}^{n} x_{i} }{n}\right)^{2} \right] \,+\,nf\left[ \frac{{\sum}_{i=1}^{n} {x_{i}^{2}} }{n}\,-\,\left( \frac{{\sum}_{i=1}^{n} x_{i} }{n}\right)^{2} \right] \,+\,n\beta \end{array} $$

The last expression describes an ideal point model with ideal points ${\sum }_{i=1}^{n} x_{i}/n$.

1.2 A.2 Proof of Lemma 1

Let a and b denote the elements of a pair of n-dimensional vectors R _n . Suppose that the inner vector products are standard spaces. By the Cauchy-Schwartz inequality, we have

$$\left( \sum\limits_{i=1}^{n} a_{i} b_{i} \right)^{2} <\left( \sum\limits_{i=1}^{n} {a_{i}^{2}} \right) \left( \sum\limits_{i=1}^{n} {b_{i}^{2}} \right) $$

Setting a _i = x _i and b _i = 1/n, we obtain

$$\left( \sum\limits_{i=1}^{n} x_{i} \frac{1}{n} \right)^{2} < \sum\limits_{i=1}^{n} {x_{i}^{2}} \sum\limits_{i=1}^{n} \frac{1}{n^{2}} \Leftrightarrow \frac{({\sum}_{i=1}^{n} x_{i} )^{2}}{n^{2}} <\frac{{\sum}_{i=1}^{n} {x_{i}^{2}} }{n} \Leftrightarrow \frac{{\sum}_{i=1}^{n} {x_{i}^{2}} }{n}-\frac{({\sum}_{i=1}^{n} x_{i} )^{2}}{n^{2}} >0, $$

which implies positive arguments of function f of the constant terms.

1.3 A.3 Proof of Proposition 2

We first consider the arguments of the similarity function f. When x _n+1 is added, the arguments change in the following amounts:

$$\frac{{\sum}_{i=1}^{n} {x_{i}^{2}} +x_{n+1}^{2}}{n+1} -\left( \frac{{\sum}_{i=1}^{n} x_{i} +x_{n+1}}{n+1}\right)^{2} -\left[ \frac{{\sum}_{i=1}^{n} {x_{i}^{2}} }{n} -\left( \frac{{\sum}_{i=1}^{n} x_{i} }{n} \right)^{2} \right] $$

Setting $a^{2}={\sum }_{i=1}^{n} {x_{i}^{2}} (\ge 0)$ and $b={\sum }_{i=1}^{n} x_{i} (\ge 0)$, this equation can be rewritten as follows:

$$\begin{array}{@{}rcl@{}} \frac{a^{2} +x_{n+1}^{2}}{n+1} &-& \left( \frac{b+x_{n+1}}{n+1}\right)^{2} -\left[ \frac{a^{2}}{n}-\left( \frac{b}{n} \right)^{2} \right] = \frac{a^{2} +x_{n+1}^{2}}{n+1}-\frac{a^{2}}{n} + \left[ -\left( \frac{b+x_{n+1}}{n+1} \right)^{2}+\left( \frac{b}{n}\right)^{2} \right] \\ &=& \frac{n(a^{2} +x_{n+1}^{2}) -(n+1)a^{2}}{n(n+1)} +\frac{(n+1)^{2} b^{2} -n^{2} (b+x_{n+1})^{2}}{n^{2} (n+1)^{2} } \\ &=& \frac{-a^{2} n(1+n)+b^{2} (1+n)+b^{2} n-2bn^{2} x_{n+1}+n^{3} x_{n+1}^{2}}{n^{2} (n+1)^{2} } \\ &=& \frac{-a^{2} n(1+n)+b^{2} (1+n)+b^{2} n+n^{3} \left( x_{n+1}-b/n\right)^{2} -b^{2} n}{n^{2} (n+1)^{2} } \\ &= &\frac{n^{3} \left( x_{n+1}-b/n\right)^{2} -(1+n)(na^{2}-b^{2})}{n^{2} (n+1)^{2} }. \end{array} $$

The sufficient condition for positivity of this expression is

$$n^{3} \left( x_{n+1}-b/n\right)^{2} -(1+n)(na^{2}+b^{2})\ge 0. $$

Conversely, the sufficient condition for negativity is

$$n^{3} \left( x_{n+1}-b/n\right)^{2} -(1+n)(na^{2}+b^{2})\le 0. $$

The inequality can be rewritten as follows:

$$\begin{array}{@{}rcl@{}} &&n^{3} \left( x_{n+1}-b/n\right)^{2} -(1+n)(na^{2}+b^{2})\ge 0 \\ &\Leftrightarrow & \ x_{n+1} \le \frac{b}{n}-\sqrt{ \frac{(1+n)(na^{2}+b^{2}) }{n^{3}} } \ \text{or} \ x_{n+1} \ge \frac{b}{n} +\sqrt{ \frac{(1+n)(na^{2}+b^{2}) }{n^{3}} }. \end{array} $$

If the above condition is satisfied, the constant term is decreasing because the above expression is increasing. On the contrary, if the following condition is satisfied, the constant term is increasing since the above expression is decreasing:

$$\begin{array}{@{}rcl@{}} &&n^{3} \left( x_{n+1}-b/n\right)^{2} -(1+n)(na^{2}+b^{2})\le 0 \\ &\Leftrightarrow & \ \frac{b}{n}-\sqrt{ \frac{(1+n)(na^{2}+b^{2}) }{n^{3}} } \le x_{n+1} \le \frac{b}{n} +\sqrt{ \frac{(1+n)(na^{2}+b^{2}) }{n^{3}} } . \end{array} $$

Note that the lower bound may be negative or greater than one. Thus, we impose two additional conditions on the operators; namely, that min{…, 1} and max{…, 0} are restrained within the interior.

When x _n+1 is added to the ideal point, we have

$$\frac{{\sum}_{i=1}^{n} x_{i} \,+\,x_{n+1}}{n+1} - \frac{{\sum}_{i=1}^{n} x_{i}}{n} \,=\,\frac{n{\sum}_{i=1}^{n} x_{i} +nx_{n+1} - (n+1) {\sum}_{i=1}^{n} x_{i} }{n(n+1) } \,=\, \frac{x_{n+1} \,-\,{\sum}_{i=1}^{n} x_{i} /n}{n+1} $$

If ${\sum }_{i=1}^{n} x_{i} /n\le x_{n+1}$ since $[ x_{n+1} -({\sum }_{i=1}^{n} x_{i} )/n] /(n+1)\ge 0$, the ideal point is increasing. On the other hand, if $x_{n+1} \le {\sum }_{i=1}^{n} x_{i} /n$, we have $[ x_{n+1} -({\sum }_{i=1}^{n} x_{i} )/n] /(n+1)\le 0$, and the ideal point is decreasing.

1.4 A.4 Proof of Proposition 3

From the assumptions of the similarity function, we have

$$\begin{array}{@{}rcl@{}} \sum\limits_{i=1}^{n}& &S(X,X_{i} )\!=\!\sum\limits_{i=1}^{n} \left[ \sum\limits_{j=1}^{m} f_{j} (x_{j}\!-\!x_{i,j} )^{2} \!+\!\sum\limits_{j=1}^{m} \beta_{j} \right] \!=\!\sum\limits_{j=1}^{m} \left[ \sum\limits_{i=1}^{n} f_{j} \left( x_{j}\!-\!x_{i,j} \right)^{2} \right] \!+\!n\left( \sum\limits_{j=1}^{m} \beta_{j} \right) \\ &\!=\!& \sum\limits_{j=1}^{m} \left\{ n f_{j} \left[ \left( x_{j}\!-\!\frac{\sum_(i\!=\!1)^{n} x_{j,i} ) }{n}\right)^{2} \right] \!+\!n f_{j} \left[ \frac{\sum_{i=1}^{n} x_{j,i}^{2} }{n} \!-\!\left( \frac{\sum_{i=1}^{n} x_{j,i} }{n}\right)^{2} \right]\right\} \!+\!n\sum\limits_{j=1}^{m} \beta_{j} \\ &\!=\!& n \sum\limits_{j=1}^{m} f_{j} \left[ \left( x_{j}\!-\!\frac{\sum_(i\!=\!1)^{n} x_{j,i} ) }{n}\right)^{2} \right] \!+\!n \sum\limits_{j=1}^{m} f_{j} \left[ \frac{\sum_{i=1}^{n} x_{j,i}^{2} }{n} \!-\!\left( \frac{\sum_{i=1}^{n} x_{j,i} }{n}\right)^{2} \right] \!+\!n\sum\limits_{j=1}^{m} \beta_{j} \end{array} $$

which expresses an ideal point model with an m-dimensional ideal points $([{\sum }_{i=1}^{n} x_{1,j} u_{i}] /[n {\sum }_{i=1}^{n} u_{i}], \ldots , [{\sum }_{i=1}^{n} x_{m,j} u_{i}] /[n {\sum }_{i=1}^{n} u_{i}] )$.

1.5 A. 5 Proof of Proposition 4

The following calculation gives

$$\begin{array}{@{}rcl@{}} && \sum\limits_{i=1}^{n} S(X,X_{i} ) u_{i} \!= \! \sum\limits_{i=1}^{n} \left\{ \left( \sum\limits_{j=1}^{m} f_{j} [(x_{j} \!- \!x_{i,j} )^{2} ] \!+ \!\sum\limits_{j=1}^{m} \beta_{j} \right) u_{i} \right\} \!= \! \sum\limits_{j=1}^{m} \left\{ \sum\limits_{i=1}^{n} f_{j} [(x_{j} \!- \!x_{i,j} )^{2} ] u_{i} \right\} \!+ \!\sum\limits_{i=1}^{n} u_{i} \sum\limits_{j=1}^{m} \beta_{j} \\ & \!=& \!n \sum\limits_{j=1}^{m}\left\{ f_{j} \left[ \left( x_{j} \!- \!\frac{{\sum}_{i=1}^{n} x_{j,i} u_{i}}{n{\sum}_{i=1}^{n} u_{i} }\right)^{2} \right] \!+ \! f_{j} \left[ \frac{{\sum}_{i=1}^{n} x_{j,i}^{2} u_{i}}{n} \!- \!\left( \frac{{\sum}_{i=1}^{n}(x_{j,i} ) u_{i}}{n{\sum}_{i=1}^{n} u_{i} }\right)^{2} \right]\right\} \!+ \!\sum\limits_{i=1}^{n} u_{i} \sum\limits_{j=1}^{m} \beta_{j} \\ & \!= \!&n\sum\limits_{j=1}^{m} f_{j} \left[ x_{j} \!- \!\left( \frac{{\sum}_{i=1}^{n} (x_{j,i} ) u_{i}}{n{\sum}_{i=1}^{n} u_{i}} \right)^{2} \right] \!+ \!n \sum\limits_{j=1}^{m} f_{j} \left[ \frac{{\sum}_{i=1}^{n} x_{j,i}^{2} u_{i}}{n} \!- \!\left( \frac{{\sum}_{i=1}^{n} (x_{j,i} ) u_{i}}{n{\sum}_{i=1}^{n} u_{i} }\right)^{2}\right] \!+ \!\sum\limits_{i=1}^{n} u_{i} \sum\limits_{j=1}^{m} \beta_{j} \end{array} $$

One can verify that the above equation corresponds to the ideal point model based on m-dimensional vectors of ideal points $([{\sum }_{i=1}^{n} x_{1,i} u_{i} ]/[n {\sum }_{i=1}^{n} u_{i}] ,\ldots , [{\sum }_{i=1}^{n} x_{m,i} u_{i}]/[n{\sum }_{i=1}^{n} u_{i}])$.

Appendix B

1.1 B.1 Discussion on the similarity function

One can calculate the weight of the similarity as either positive or negative. To do so, let K ⁺ and K ⁻ denote a set of cases with k _h,j ≥ 0 and k _h,j < 0, respectively, among m attributes. Classifying two cases, we have

$$S_{h} (X,X_{i} )=\sum\limits_{j=1}^{m} k_{h,j} s_{j} (x_{j},x_{i,j} ) I_{K^{+}} (k)+\sum\limits_{j=1}^{m} k_{h,j} s_{j} (x_{j},x_{i,j} ) I_{K^{-}} (k), $$

where $I_{K^{+}} (k)$ and $I_{K^{-}} (k)$ indicate functions of positive and negative cases, respectively. Then, since a part of k _h,j s _j (x _j ,x _i,j ) > 0 of the first term ${\sum }_{j=1}^{m} k_{h,j} s_{j} (x_{j},x_{i,j} ) I_{K^{+}} (k)$ is considered as one kind of similarity function, k _h,j ≥ 0 can be transformed into an ideal point. However, for the second term ${\sum }_{j=1}^{m} k_{h,j} s_{j} (x_{j},x_{i,j} ) I_{K^{-}} (k)$, setting $-{\sum }_{j=1}^{m} (-k_{h,j})s_{j} (x_{j},x_{i,j} ) I_{K^{-}} (k)$, since a part of (−k _h,j )s _j (x _j ,x _i,j ) > 0 becomes one kind of similarity function, and k _h,j < 0 can also be transformed into another ideal point.

1.2 B.2 Our estimation method

Based on Rossi et al. (2005), our parameters are estimated using the following method (see Rossi et al. (2005) for details). Given data y, the simultaneous posterior distribution is given by

$$ p(k_{h},{\Delta}, V_{k}| y)\propto p({\Delta},V_{k}){\Pi}_{h=1}^{H} \{ p(k_{h} |{\Delta}, V_{k},z_{h} )p(y_{h} |k_{h},x_{h})\}. $$

(3)

Thus, conditional posterior distribution p(k _h |Δ,V _k ,z _h ,y _h ,x _h ) is

$$p(k_{h}| {\Delta}, V_{k},z_{h},y_{h},x_{h})\propto p(y_{k}| k_{h},x_{h}) p(k_{h} |{\Delta}, V_{k},z_{h} ). $$

However, since conjugate distribution cannot be explicitly calculated, sampling according to a random walk algorithm is conducted. Let ${k^{0}_{h}}$ denote an initial value. Then, the value is chosen as ${k_{h}^{t}}=k_{h}^{t-1}+\varepsilon ^{\prime }$, where t > 0 is an index and ε follows a normal distribution. If a sampling number chosen by a uniform random number U D _[0,1] is below an acceptance rate (4), the next ${k_{h}^{t}}$ is determined.

$$ \alpha (k_{h}^{t-1}; {k_{h}^{t}})=\min \{ p({k_{h}^{t}} |{\Delta} ,V_{k},z_{h},y_{h},x_{h})/ p(k_{h}^{t-1}| {\Delta} , V_{k}, z_{h},y_{h},x_{h} ),1 \} $$

(4)

A conditional distribution on V _k can be sampled as follows

$$\begin{array}{@{}rcl@{}} vec ({\Delta}) &&\sim N_{k}q (vec(\tilde{D} )| V_{k} \otimes (Z^{\prime} Z+A_{d} )^{-1} ) \\ \tilde{D} &=& (Z^{\prime}+A_{d})^{-1} (Z^{\prime} Z((Z^{\prime} Z)^{-1} Z^{\prime}K)+A_{d})^{-1}, \end{array} $$

where conditional distribution on Δ is a vector-valued function of vec, Z is a matrix combined with attributes z _h , and K is a matrix with all k _h . A _d is a variance of a prior distribution assumed to be a normal distribution.

Conditional distribution on Δ can be sampled, and follows the next equations:

$$\begin{array}{@{}rcl@{}} V_{k} &&\sim IW(f_{0} +H, V_{0}+V^{\prime}) \\ V^{\prime} &=&\sum\limits_{h=1}^{H} (k_{h}-{\Delta} 'z_{h} ) (k_{h}-{\Delta} 'z_{h})', \end{array} $$

where f ₀ and V ₀ are the initial parameters of inverse Wishart distribution.