CQR/CER with multiple outputs

Similarly to CNLS with DDF, we present another two approaches integrating DDF to convex quantile/expectile regression.

• CQR-DDF model

  \begin{alignat}{2} \underset{\alpha,\boldsymbol{\beta},\boldsymbol{\gamma},\varepsilon^{+},\varepsilon^{-}}{\mathop{\min }}&\, \tau \sum\limits_{i=1}^{n}{\varepsilon _{i}^{+}}+(1-\tau )\sum\limits_{i=1}^{n}{\varepsilon _{i}^{-}} &{\quad}& \\ \textit{s.t.}\quad & \boldsymbol{\gamma}_i^{'}\boldsymbol{y}_i = \alpha_i + \boldsymbol{\beta}_i^{'}\boldsymbol{x}_i + \varepsilon^+_i - \varepsilon^-_i &{\quad}& \forall i \notag \\ & \alpha_i + \boldsymbol{\beta}_i^{'}\boldsymbol{x}_i -\boldsymbol{\gamma}_i^{'}\boldsymbol{y}_i \le \alpha_j + \boldsymbol{\beta}_j^{'}\boldsymbol{x}_i -\boldsymbol{\gamma}_j^{'}\boldsymbol{y}_i &{\quad}& \forall i, j \notag \\ & \boldsymbol{\gamma}_i^{'} g^{y} + \boldsymbol{\beta}_i^{'} g^{x} = 1 &{\quad}& \forall i \notag \\ & \boldsymbol{\beta}_i \ge \boldsymbol{0} , \boldsymbol{\gamma}_i \ge \boldsymbol{0} &{\quad}& \forall i \notag \\ & \varepsilon _i^{+}\ge 0,\ \varepsilon_i^{-} \ge 0 &{\quad}& \forall i \notag \end{alignat}

• CER-DDF model

  \begin{alignat}{2} \underset{\alpha,\boldsymbol{\beta},\boldsymbol{\gamma},\varepsilon^{+},\varepsilon^{-}}{\mathop{\min}}&\, \tilde{\tau} \sum\limits_{i=1}^n(\varepsilon _i^{+})^2+(1-\tilde{\tau} )\sum\limits_{i=1}^n(\varepsilon_i^{-})^2 &{\quad}& \\ \textit{s.t.}\quad & \boldsymbol{\gamma}_i^{'}\boldsymbol{y}_i = \alpha_i + \boldsymbol{\beta}_i^{'}\boldsymbol{x}_i + \varepsilon^+_i - \varepsilon^-_i &{\quad}& \forall i \notag \\ & \alpha_i + \boldsymbol{\beta}_i^{'}\boldsymbol{x}_i -\boldsymbol{\gamma}_i^{'}\boldsymbol{y}_i \le \alpha_j + \boldsymbol{\beta}_j^{'}\boldsymbol{x}_i -\boldsymbol{\gamma}_j^{'}\boldsymbol{y}_i &{\quad}& \forall i, j \notag \\ & \boldsymbol{\gamma}_i^{'} g^{y} + \boldsymbol{\beta}_i^{'} g^{x} = 1 &{\quad}& \forall i \notag \\ & \boldsymbol{\beta}_i \ge \boldsymbol{0} , \boldsymbol{\gamma}_i \ge \boldsymbol{0} &{\quad}& \forall i \notag \\ & \varepsilon _i^{+}\ge 0,\ \varepsilon_i^{-} \ge 0 &{\quad}& \forall i \notag \end{alignat}

Example: CQR-DDF [.ipynb]

# import packages
from pystoned import CQERDDF
from pystoned.constant import FUN_PROD, OPT_LOCAL
from pystoned import dataset as dataset

# import Finnish electricity distribution firms data
data = dataset.load_Finnish_electricity_firm(x_select=['OPEX', 'CAPEX'],
                                    y_select=['Energy', 'Length', 'Customers'])

# define and solve the CQR-DDF model
model = CQERDDF.CQRDDF(y=data.y, x=data.x, b=None, tau=0.9, fun = FUN_PROD, gx= [1.0, 0.0], gb=None, gy= [0.0, 0.0, 0.0])
model.optimize(OPT_LOCAL)

# display the residual
model.display_positive_residual()
model.display_negative_residual()