Distinguish between type 1 and 2 conditional biases

Author: dnerini

pysteps/tests/test_detcontscores.py

@@ -40,8 +40,10 @@
     (fct_data, obs_data, ('NMSE'), None, None, [0.113711]),
     # Root Mean Square Error
     (fct_data, obs_data, ('RMSE'), None, None, [8.009370]),
-    # Beta
-    (fct_data, obs_data, ('beta'), None, None, [0.498200]),
+    # Beta1
+    (fct_data, obs_data, ('beta1'), None, None, [0.498200]),
+    # Beta2
+    (fct_data, obs_data, ('beta2'), None, None, [0.591673]),
     # reduction of variance
     (fct_data, obs_data, ('RV'), None, None, [-0.054622]),
     # debiased RMSE

pysteps/verification/detcontscores.py

@@ -40,7 +40,9 @@ def det_cont_fct(pred, obs, scores="", axis=None, conditioning=None, thr=0.0):
         +------------+--------------------------------------------------------+
         | Name       | Description                                            |
         +============+========================================================+
-        |  beta      | linear regression slope (conditional bias)             |
+        |  beta1     | linear regression slope (type 1 conditional bias)      |
+        +------------+--------------------------------------------------------+
+        |  beta2     | linear regression slope (type 2 conditional bias)      |
         +------------+--------------------------------------------------------+
         |  corr_p    | pearson's correleation coefficien (linear correlation) |
         +------------+--------------------------------------------------------+
@@ -98,6 +100,12 @@ def det_cont_fct(pred, obs, scores="", axis=None, conditioning=None, thr=0.0):
     Note that "scatter" is the only score that will be computed in dB units of
     the multiplicative error, i.e.: 10*log10(pred/obs).
 
+    beta1 measures the degree of conditional bias of the observations given the
+    forecasts (type 1).
+
+    beta2 measures the degree of conditional bias of the forecasts given the
+    observations (type 2).
+
     The normalized MSE is computed as NMSE = E[(pred - obs)^2]/E[(pred + obs)^2].
 
     The debiased RMSE is computed as DRMSE = sqrt(RMSE - ME^2).
@@ -116,6 +124,10 @@ def det_cont_fct(pred, obs, scores="", axis=None, conditioning=None, thr=0.0):
     precipitation measurement in a mountainous region. Q.J.R. Meteorol. Soc.,
     132: 1669-1692. doi:10.1256/qj.05.190
 
+    Potts, J. (2012), Chapter 2 - Basic concepts. Forecast verification: a
+    practitioner’s guide in atmospheric sciences, I. T. Jolliffe, and D. B.
+    Stephenson, Eds., Wiley-Blackwell, 11–29.
+
     See also
     --------
 
@@ -420,7 +432,9 @@ def det_cont_fct_compute(err, scores=""):
         +------------+--------------------------------------------------------+
         | Name       | Description                                            |
         +============+========================================================+
-        |  beta      | linear regression slope (conditional bias)             |
+        |  beta1      | linear regression slope (type 1 conditional bias)     |
+        +------------+--------------------------------------------------------+
+        |  beta2      | linear regression slope (type 2 conditional bias)     |
         +------------+--------------------------------------------------------+
         |  corr_p    | pearson's correleation coefficien (linear correlation) |
         +------------+--------------------------------------------------------+
@@ -496,10 +510,15 @@ def get_iterable(x):
             corr_p = err["cov"] / np.sqrt(err["vobs"]) / np.sqrt(err["vpred"])
             result["corr_p"] = corr_p
 
-        # beta (linear regression slope)
-        if score_ in ["beta", ""]:
-            beta = err["cov"] / err["vpred"]
-            result["beta"] = beta
+        # beta1 (linear regression slope)
+        if score_ in ["beta", "beta1", ""]:
+            beta1 = err["cov"] / err["vpred"]
+            result["beta1"] = beta1
+
+        # beta2 (linear regression slope)
+        if score_ in ["beta2", ""]:
+            beta2 = err["cov"] / err["vobs"]
+            result["beta2"] = beta2
 
         # debiased RMSE
         if score_ in ["drmse", ""]:

pysteps/verification/interface.py

@@ -53,7 +53,9 @@ def get_method(name, type="deterministic"):
         +------------+--------------------------------------------------------+
         |  SEDI      | symmetric extremal dependency index                    |
         +------------+--------------------------------------------------------+
-        |  beta      | linear regression slope (conditional bias)             |
+        |  beta1     | linear regression slope (type 1 conditional bias)      |
+        +------------+--------------------------------------------------------+
+        |  beta2     | linear regression slope (type 2 conditional bias)      |
         +------------+--------------------------------------------------------+
         |  corr_p    | pearson's correleation coefficien (linear correlation) |
         +------------+--------------------------------------------------------+
@@ -120,25 +122,35 @@ def get_method(name, type="deterministic"):
 
     Multiplicative scores can be computed by passing log-tranformed values.
     Note that "scatter" is the only score that will be computed in dB units of
-    the multiplicative error, i.e.: 10log10(pred/obs).
+    the multiplicative error, i.e.: 10*log10(pred/obs).
+
+    beta1 measures the degree of conditional bias of the observations given the
+    forecasts (type 1).
+
+    beta2 measures the degree of conditional bias of the forecasts given the
+    observations (type 2).
 
-    The debiased RMSE is computed as DRMSE = sqrt(RMSE - ME^2)
+    The normalized MSE is computed as NMSE = E[(pred - obs)^2]/E[(pred + obs)^2].
 
-    The reduction of variance score is computed as RV = 1 - MSE/Var(obs)
+    The debiased RMSE is computed as DRMSE = sqrt(RMSE - ME^2).
+
+    The reduction of variance score is computed as RV = 1 - MSE/Var(obs).
 
     Score names denoted by * can only be computed offline, meaning that the
-    these cannot be update using _init, _accum and _compute methods of this
+    these cannot be computed using _init, _accum and _compute methods of this
     module.
 
-    Score names denoted by * can only be computed offline.
-
     References
     ----------
 
     Germann, U. , Galli, G. , Boscacci, M. and Bolliger, M. (2006), Radar
     precipitation measurement in a mountainous region. Q.J.R. Meteorol. Soc.,
     132: 1669-1692. doi:10.1256/qj.05.190
 
+    Potts, J. (2012), Chapter 2 - Basic concepts. Forecast verification: a
+    practitioner’s guide in atmospheric sciences, I. T. Jolliffe, and D. B.
+    Stephenson, Eds., Wiley-Blackwell, 11–29.
+
     """
 
     if name is None:
@@ -178,6 +190,8 @@ def f(fct, obs, **kwargs):
         # continuous
         elif name in [
             "beta",
+            "beta1",
+            "beta2",
             "corr_p",
             "corr_s",
             "drmse",