Adjust the DQM formula (#107)

cmethods/distribution.py

@@ -144,12 +144,8 @@ def detrended_quantile_mapping(
         m_simp_mean = np.nanmean(m_simp)
 
         if kind in ADDITIVE:
-            epsilon = np.interp(
-                m_simp - m_simp_mean + m_simh_mean,
-                xbins,
-                cdf_simh,
-            )  # Eq. 1
-            X = get_inverse_of_cdf(cdf_obs, epsilon, xbins) + m_simp_mean - m_simh_mean  # Eq. 1
+            epsilon = np.interp(m_simp - m_simp_mean, xbins, cdf_simh)  # Eq. 1
+            X = get_inverse_of_cdf(cdf_obs, epsilon, xbins) + m_simp_mean  # Eq. 1
 
         else:  # kind in cls.MULTIPLICATIVE:
             epsilon = np.interp(  # Eq. 2

doc/methods.rst

@@ -55,6 +55,12 @@ for both additive and multiplicative Linear Scaling are shown:
 
         X^{*LS}_{sim,p}(i) = X_{sim,p}(i) + \mu_{m}(X_{obs,h}(i)) - \mu_{m}(X_{sim,h}(i))
 
+where:
+
+    .. math::
+
+        \mu_m(X\ldots(i)) = \text{long-term monthly mean of month related to day at index } i
+
 **Multiplicative**:
 
     The multiplicative Linear Scaling differs from the additive variant in such
@@ -64,6 +70,12 @@ for both additive and multiplicative Linear Scaling are shown:
 
         X^{*LS}_{sim,h}(i) = X_{sim,h}(i) \cdot \left[\frac{\mu_{m}(X_{obs,h}(i))}{\mu_{m}(X_{sim,h}(i))}\right]
 
+where:
+
+    .. math::
+
+        \mu_m(X\ldots(i)) = \text{long-term monthly mean of month related to day at index } i
+
 
 .. code-block:: python
     :linenos:
@@ -119,6 +131,12 @@ deviation in the mean.
 
     X^{*LS}_{sim,p}(i) = X_{sim,p}(i) + \mu_{m}(X_{obs,h}(i)) - \mu_{m}(X_{sim,h}(i))
 
+where:
+
+    .. math::
+
+        \mu_m(X\ldots(i)) = \text{long-term monthly mean of month related to day at index } i
+
 **(2)** In the second step, the time-series are shifted to a zero mean. This
 enables the adjustment of the standard deviation in the following step.
 
@@ -202,6 +220,12 @@ for both additive and multiplicative Delta Method are shown:
 
         X^{*DM}_{sim,p}(i) = X_{obs,h}(i) + \mu_{m}(X_{sim,p}(i)) - \mu_{m}(X_{sim,h}(i))
 
+where:
+
+    .. math::
+
+        \mu_m(X\ldots(i)) = \text{long-term monthly mean of month related to day at index } i
+
 **Multiplicative**:
 
     The multiplicative variant behaves like the additive, but with the
@@ -212,6 +236,12 @@ for both additive and multiplicative Delta Method are shown:
 
         X^{*DM}_{sim,p}(i) = X_{obs,h}(i) \cdot \left[\frac{ \mu_{m}(X_{sim,p}(i)) }{ \mu_{m}(X_{sim,h}(i))}\right]
 
+where:
+
+    .. math::
+
+        \mu_m(X\ldots(i)) = \text{long-term monthly mean of month related to day at index } i
+
 .. code-block:: python
     :linenos:
     :caption: Example: Delta Method
@@ -325,10 +355,11 @@ Detrended Quantile Mapping
 The Detrended Quantile Mapping bias correction technique can be used to minimize
 distributional biases between modeled and observed time-series climate data like
 the regular Quantile Mapping. Detrending means, that the values of
-:math:`X_{sim,p}` are shifted by the mean of :math:`X_{sim,h}` before the
-regular Quantile Mapping is applied. After the Quantile Mapping was applied, the
-mean is shifted back. Since it does not make sense to take the whole mean to
-rescale the data, the month-dependent long-term mean is used.
+:math:`X_{sim,p}` are shifted by the mean of :math:`X_{sim,p}` before the
+regular Quantile Mapping is applied. The shift is performed on a monthly basis.
+After the Quantile Mapping was applied, the mean is shifted back. Since it does
+not make sense to take the whole mean to rescale the data, the month-dependent
+long-term mean is used.
 
 This method must be applied on a 1-dimensional data set i.e., there is only one
 time-series passed for each of ``obs``, ``simh``, and ``simp``. This method
@@ -351,17 +382,26 @@ shift of :math:`X_{sim,p}(i)`:
 
     .. math::
 
-        X_{sim,p}^{*DT}(i) & = X_{sim,p}(i) + \Delta\mu \\[1pt]
+        X_{sim,p}^{*DT}(i) & = X_{sim,p}(i) - \mu_m(X_{sim,p}(i)) \\[1pt]
         X_{sim,p}^{*DQM}(i) & = F_{obs,h}^{-1}\left\{F_{sim,h}\left[X_{sim,p}^{*DT}(i)\right]\right\}
 
+where:
+
+    .. math::
+
+        \mu_m(X\ldots(i)) = \text{long-term monthly mean of month related to day at index } i
 
 **Multiplicative**:
 
     .. math::
 
-        X_{sim,p}^{*DT}(i) & = X_{sim,p}(i) \cdot \Delta\mu \\[1pt]
-        X^{*DQM}_{sim,p}(i) & = F^{-1}_{obs,h}\Biggl\{F_{sim,h}\left[\frac{\mu{X_{sim,h}} \cdot X_{sim,p}^{*DT}(i)}{\mu{X_{sim,p}^{*DT}(i)}}\right]\Biggr\}\frac{\mu{X_{sim,p}^{*DT}(i)}}{\mu{X_{sim,h}}}
+        X^{*DQM}_{sim,p}(i) = F^{-1}_{obs,h}\Biggl\{F_{sim,h}\left[\frac{\mu_m(X_{sim,h}(i)) \cdot X_{sim,p}(i)}{\mu_m(X_{sim,p}(i))}\right]\Biggr\}\frac{\mu_m(X_{sim,p}(i))}{\mu_m(X_{sim,h}(i))}
+
+where:
+
+    .. math::
 
+        \mu_m(X\ldots(i)) = \text{long-term monthly mean of month related to day at index } i
 
 .. code-block:: python
     :linenos: