Add support for Gamma regression (#1258)

* Add support for Gamma regression

* Use base_score to replace the lp_bias

* Remove the lp_bias config block

* Add a demo for running gamma regression in Python

* Typo fix

* Revise the description for objective

* Add a script to generate the autoclaims dataset

src/metric/elementwise_metric.cc

@@ -155,5 +155,87 @@ XGBOOST_REGISTER_METRIC(PossionNegLoglik, "poisson-nloglik")
 .describe("Negative loglikelihood for poisson regression.")
 .set_body([](const char* param) { return new EvalPoissionNegLogLik(); });
 
+/*!
+ * \brief base class of element-wise evaluation
+ *      with additonal dispersion parameter
+ * \tparam Derived the name of subclass
+ */
+template<typename Derived>
+struct EvalEWiseBase2 : public Metric {
+  float Eval(const std::vector<float>& preds,
+             const MetaInfo& info,
+             bool distributed) const override {
+    CHECK_NE(info.labels.size(), 0) << "label set cannot be empty";
+    CHECK_EQ(preds.size(), info.labels.size())
+        << "label and prediction size not match, "
+        << "hint: use merror or mlogloss for multi-class classification";
+    const omp_ulong ndata = static_cast<omp_ulong>(info.labels.size());
+
+    // Computer dispersion
+    double sum = 0.0, wsum = 0.0;
+    #pragma omp parallel for schedule(static)
+    for (omp_ulong i = 0; i < ndata; ++i) {
+      const float wt = info.GetWeight(i);
+      sum += static_cast<const Derived*>(this)->EvalDispersion(info.labels[i], preds[i]) * wt;
+      wsum += wt;
+    }
+    double dat[2]; dat[0] = sum, dat[1] = wsum;
+    if (distributed) {
+      rabit::Allreduce<rabit::op::Sum>(dat, 2);
+    }
+    double dispersion = dat[0] / (dat[1] - info.num_col);
+
+    // Computer metric
+    sum = 0.0, wsum = 0.0;
+    #pragma omp parallel for reduction(+: sum, wsum) schedule(static)
+    for (omp_ulong i = 0; i < ndata; ++i) {
+      const float wt = info.GetWeight(i);
+      sum += static_cast<const Derived*>(this)->EvalRow(info.labels[i], preds[i], dispersion) * wt;
+      wsum += wt;
+    }
+    dat[0] = sum, dat[1] = wsum;
+    if (distributed) {
+      rabit::Allreduce<rabit::op::Sum>(dat, 2);
+    }
+    return Derived::GetFinal(dat[0], dat[1]);
+  }
+  /*!
+   * \brief to be implemented by subclass,
+   *   get evaluation result from one row
+   * \param label label of current instance
+   * \param pred prediction value of current instance
+   */
+  inline float EvalRow(float label, float pred, float dispersion) const;
+  /*!
+   * \brief to be overridden by subclass, final transformation
+   * \param esum the sum statistics returned by EvalRow
+   * \param wsum sum of weight
+   */
+  inline static float GetFinal(float esum, float wsum) {
+    return esum / wsum;
+  }
+  inline float EvalDispersion(float label, float pred) const;
+};
+
+struct EvalGammaNegLogLik : public EvalEWiseBase2<EvalGammaNegLogLik> {
+  const char *Name() const override {
+    return "gamma-nloglik";
+  }
+  inline float EvalRow(float y, float py, float psi) const {
+    double theta = -1. / py;
+    double a = psi;
+    double b = -std::log(-theta);
+    double c = 1. / psi * std::log(y/psi) - std::log(y) - common::LogGamma(1. / psi);
+    return -((y * theta - b) / a + c);
+  }
+  inline float EvalDispersion(float y, float py) const {
+    return ((y - py) * (y - py)) / (py * py);
+  }
+};
+
+XGBOOST_REGISTER_METRIC(GammaNegLoglik, "gamma-nloglik")
+.describe("Negative loglikelihood for gamma regression.")
+.set_body([](const char* param) { return new EvalGammaNegLogLik(); });
+
 }  // namespace metric
 }  // namespace xgboost