0.12229 by KernelRidge

dataloader.py

@@ -2,7 +2,12 @@ import os
 import csv
 import numpy as np
 import random
-# import matplotlib.pyplot as plt
+from sklearn.decomposition import PCA 
+from sklearn.model_selection import GridSearchCV
+from sklearn.kernel_ridge import KernelRidge
+from sklearn.svm import SVR
+import evaluation
+from description_map import value_map,fix_key,fix_miss,add_future
 
 # load description_txt
 description_txt = []
@@ -14,38 +19,46 @@ for i,line in enumerate(open('./datasets/data_description.txt'),0):
     description_txt.append(line)
 colon_indexs.append(524)#the end of description
 description_length = len(colon_indexs)-1
-print('Description length:',description_length)
 
 
-
-descriptions = []
+Full_map = {}
+desc_keys = []
 for i in range(len(colon_indexs)-1):
-    mapping = {}
-    description_title = description_txt[colon_indexs[i]]
+    ori_map = {}
+    my_map = {}
+    desc_key = description_txt[colon_indexs[i]]
+    desc_key = desc_key[:desc_key.find(':')]
+    desc_keys.append(desc_key)
+    # print(desc_key)
     interspace = colon_indexs[i+1]-colon_indexs[i]-2 #two space line
+
     if interspace == 0:
-        mapping['Just_num'] = 'None'
-        descriptions.append(mapping)
+        ori_map['Just_num'] = 'None'
+        Full_map[desc_key]=ori_map
     else:
         for j in range(interspace-1): #del low space line
             line = description_txt[colon_indexs[i]+j+2]
-            mapping_key = line[:line.find('\t')]
+            key = line[:line.find('\t')]
             #data_description.txt is wrong here
-            if mapping_key == 'NA ':
-                mapping_key = 'NA'
-            if mapping_key == 'WD ':
-                mapping_key = 'WD'
-            if mapping_key == 'BrkComm' or mapping_key =='Brk Cmn':
-                mapping_key = 'BrkCmn'
-            mapping[mapping_key] = j
-        descriptions.append(mapping)
-# print(descriptions)
-
-def match_random(a,b):
-    state = np.random.get_state()
-    np.random.shuffle(a)
-    np.random.set_state(state)
-    np.random.shuffle(b)
+            if key == 'NA ':
+                key = 'NA'
+            if key == 'WD ':
+                key = 'WD'
+            if key == 'BrkComm' or key =='Brk Cmn':
+                key = 'BrkCmn'
+            if desc_key in value_map:
+                my_map[key] = value_map[desc_key][key]
+                Full_map['my_'+desc_key]=my_map
+            ori_map[key] = interspace-j-1 #change word to vector
+            Full_map[desc_key]=ori_map
+
+# def normlize(npdata,justprice = False):
+#     _mean = np.mean(npdata)
+#     _std = np.std(npdata)
+#     if justprice:       
+#         _mean = 180921.195
+#         _std = 79415.2918
+#     return (npdata-_mean)/_std
 
 def normlize(npdata,justprice = False):
     _min = np.min(npdata)
@@ -55,92 +68,124 @@ def normlize(npdata,justprice = False):
         _max = 755000.0
     return (npdata-_min)/(_max-_min)
 
+# def convert2price(tensor):
+#     return tensor*79415.2918+180921.195
+
 def convert2price(tensor):
     return tensor*(755000.0-34900.0)+34900
 
-
-def fix_key(key):
-    #csv is wrong here
-    if key == 'Wd Shng':
-        key='WdShing'
-    if key == '2fmCon':
-        key='2FmCon'
-    if key == 'NAmes':
-        key='Names'
-    if key == 'Duplex':
-        key='Duplx'
-    if key == 'CmentBd':
-        key='CemntBd'
-    if key == 'C (all)':
-        key='C'
-    if key == 'Twnhs':
-        key='TwnhsI'
-    if key == 'Brk Cmn' or key =='BrkComm':
-        key='BrkCmn'
-    else:
-        key = key
-    return key
-
 def load_train():
-    ##load train csv
+    
+    desc_map = {}
+    price_map = {}
     csv_data = []
+    #train_del_1299_524.csv
     reader = csv.reader(open('./datasets/train.csv'))
     for line in reader:
         csv_data.append(line)
     id_length = len(csv_data)-1
 
-    data = np.zeros((id_length,description_length+1))
-    for i in range(id_length):
-        for j in range(description_length+1):
-            key = csv_data[i+1][j+1]
+    for i in range(80):
+        arr = np.zeros(id_length)
+        my_arr = np.zeros(id_length)
+        for j in range(id_length):
+            key = csv_data[j+1][i+1]
             key = fix_key(key)
-            if j == description_length:
-                data[i][j] = float(key)
+            if i == 79:
+                arr[j] = float(key)
             else:
-                if key in descriptions[j]: #SalePrice
-                    data[i][j] = float(descriptions[j][key])
-                else:#just num here
-                    # print(i,j)
+                #my map
+                if desc_keys[i] in value_map:
+                    if key == 'NA':
+                        my_arr[j] = fix_miss(desc_keys[i])
+                    else:
+                        my_arr[j] = Full_map['my_'+desc_keys[i]][key]
+                #auto map
+                if key in Full_map[desc_keys[i]]:
+                    arr[j] = Full_map[desc_keys[i]][key]
+                else:
                     if key == 'NA':
-                        key = 0;
-                    data[i][j] = float(key)
-    return data
+                        arr[j] = fix_miss(desc_keys[i])
+                    else:
+                        arr[j] = float(key)
+        if i == 79:        
+            price_map['price']=arr
+        else:
+            if desc_keys[i] in value_map:
+                desc_map['my_'+desc_keys[i]] = my_arr
+            # else:
+            desc_map[desc_keys[i]] = arr
+    return desc_map,price_map
 
 def load_test():
-    ##load train csv
+    desc_map = {}
     csv_data = []
     reader = csv.reader(open('./datasets/test.csv'))
     for line in reader:
         csv_data.append(line)
     id_length = len(csv_data)-1
 
-    data = np.zeros((id_length,description_length))
-    for i in range(id_length):
-        for j in range(description_length):
-            key = csv_data[i+1][j+1]
+    for i in range(79):
+        arr = np.zeros(id_length)
+        my_arr = np.zeros(id_length)
+        for j in range(id_length):
+            key = csv_data[j+1][i+1]
             key = fix_key(key)
-            if j == description_length:
-                data[i][j] = float(key)
+
+            #my map
+            if desc_keys[i] in value_map:
+                if key == 'NA':
+                    my_arr[j] = fix_miss(desc_keys[i])
+                else:
+                    my_arr[j] = Full_map['my_'+desc_keys[i]][key]
+
+            #auto map
+            if key in Full_map[desc_keys[i]]:
+                arr[j] = Full_map[desc_keys[i]][key]
             else:
-                if key in descriptions[j]: #SalePrice
-                    data[i][j] = float(descriptions[j][key])
-                else:#just num here
-                    # print(i,j)
-                    if key == 'NA':
-                        key = 0;
-                    data[i][j] = float(key)
-    return data
-
-def load_all():
-    train_desc = load_train()[:,:79]
-    train_price = load_train()[:,79]
-    test_desc = load_test()
+                if key == 'NA':
+                    arr[j] = fix_miss(desc_keys[i])
+                else:
+                    arr[j] = float(key)
+
+        if desc_keys[i] in value_map:
+            desc_map['my_'+desc_keys[i]] = my_arr
+        # else:
+        desc_map[desc_keys[i]] = arr
+    return desc_map
+# for i,word in enumerate(wordlist,0):
+
+def dict2numpy(dict_data):
+    value_0 = list(dict_data.values())[0]
+    np_data = np.zeros((len(value_0),len(dict_data)))
+    for i,key in  enumerate(dict_data.keys(),0):
+        np_data[:,i] = np.array(dict_data[key])
+    return np_data
+
+def load_all(dimension):
+    desc_map,price_map = load_train()
+    desc_map = add_future(desc_map)
+    # print(len(desc_map))
+    # print(desc_map)
+    # print(desc_map)
+    train_price = np.array(price_map['price'])
+    train_desc = dict2numpy(desc_map)
+
+    desc_map = load_test()
+    desc_map = add_future(desc_map)
+    test_desc = dict2numpy(desc_map)
+
     desc_all = np.concatenate((train_desc,test_desc),axis=0)
-    for i in range(description_length):
+    for i in range(len(desc_all[0])):
         desc_all[:,i] = normlize(desc_all[:,i])
-    train_price = normlize(train_price)
+    # print(desc_all)
+    pca=PCA(n_components=dimension)     #加载PCA算法，设置降维后主成分数目为
+    desc_all=pca.fit_transform(desc_all)#对样本进行降维
+
+    train_price = normlize(train_price,True)
     train_desc = desc_all[:len(train_desc)]
     test_desc = desc_all[len(train_desc):]
+
     return train_desc.astype(np.float32),train_price.astype(np.float32),test_desc.astype(np.float32)
 
 def write_csv(prices,path):
@@ -152,9 +197,25 @@ def write_csv(prices,path):
     csvFile.close()
 
 def main():
-    train_desc,train_price,test_desc = load_all()
-    print(len(test_desc))
 
+    dimension = 80
+
+
+    train_desc,train_price,test_desc = load_all(dimension)
+
+    # # KRR = KernelRidge(alpha=0.6, kernel='polynomial', degree=2, coef0=2.5)
+
+    # kr = GridSearchCV(KernelRidge(kernel='polynomial', gamma=0.1),
+    #                   param_grid={"alpha": [1e0, 0.1, 1e-2, 1e-3],
+    #                               "gamma": np.logspace(-2, 2, 5)})
+
+
+    # kr.fit(train_desc, train_price)
+    # y_kr = kr.predict(test_desc)
+    # for i in range(len(y_kr)):
+    #     y_kr[i] = convert2price(y_kr[i])
+    # # print(y_kr.shape)
+    # print(dimension,evaluation.eval_test(y_kr))
 
     # write_csv(train_price, './result.csv')
     # # print(data)

description_map.py

+import numpy as np
+
+value_map = {}
+value_map["MSSubClass"] = {'180':1, 
+                            '30':2, '45':2, 
+                            '190':3, '50':3, '90':3, 
+                            '85':4, '40':4, '160':4, 
+                            '70':5, '20':5, '75':5, '80':5, '150':5,
+                            '120': 6, '60':6}
+
+value_map["MSZoning"] = {'A':1,'C':4, 'FV':1, 'I':3,'RH':3, 'RL':2, 'RP':3, 'RM':2}
+
+value_map["Neighborhood"] = {'MeadowV':1,
+                               'IDOTRR':2, 'BrDale':2,
+                               'OldTown':3, 'Edwards':3, 'BrkSide':3,
+                               'Sawyer':4, 'Blueste':4, 'SWISU':4, 'Names':4,
+                               'NPkVill':5, 'Mitchel':5,
+                               'SawyerW':6, 'Gilbert':6, 'NWAmes':6,
+                               'Blmngtn':7, 'CollgCr':7, 'ClearCr':7, 'Crawfor':7,
+                               'Veenker':8, 'Somerst':8, 'Timber':8,
+                               'StoneBr':9,
+                               'NoRidge':10, 'NridgHt':10}
+
+value_map["Condition1"] = {'Artery':1,
+                           'Feedr':2, 'RRAe':2,
+                           'Norm':3, 'RRAn':3,
+                           'PosN':4, 'RRNe':4,
+                           'PosA':5 ,'RRNn':5}
+
+value_map["BldgType"] = {'2FmCon':1, 'Duplx':1, 'TwnhsI':1, '1Fam':2, 'TwnhsE':2}
+
+value_map["HouseStyle"] = {'1.5Unf':1, 
+                           '1.5Fin':2, '2.5Unf':2, 'SFoyer':2, 
+                           '1Story':3, 'SLvl':3,
+                           '2Story':4, '2.5Fin':4}
+
+value_map["Exterior1st"] = {'BrkCmn':1,
+                             'AsphShn':2, 'CBlock':2, 'AsbShng':2,
+                             'WdShing':3, 'Wd Sdng':3, 'MetalSd':3, 'Stucco':3, 'HdBoard':3,'Other':3,
+                             'BrkFace':4, 'Plywood':4, 'PreCast':4,
+                             'VinylSd':5,
+                             'CemntBd':6,
+                             'Stone':7, 'ImStucc':7}
+
+value_map["MasVnrType"] = {'BrkCmn':1, 'None':1, 'CBlock':1,'BrkFace':2, 'Stone':3}
+
+value_map["ExterQual"] = {'Po':1,'Fa':2, 'TA':3, 'Gd':4, 'Ex':5}
+
+value_map["Foundation"] = {'Slab':1, 
+                           'BrkTil':2, 'CBlock':2, 'Stone':2,
+                           'Wood':3, 'PConc':4}
+
+value_map["BsmtQual"] = {'NA':1, 'Po':2,'Fa':3, 'TA':4, 'Gd':5, 'Ex':6}
+
+value_map["BsmtExposure"] = {'NA':1, 'No':2, 'Av':3, 'Mn':3, 'Gd':4}
+
+value_map["Heating"] = {'Floor':1, 'Grav':1, 'Wall':2, 'OthW':3, 'GasW':4, 'GasA':5}
+
+value_map["HeatingQC"] = {'Po':1, 'Fa':2, 'TA':3, 'Gd':4, 'Ex':5}
+
+value_map["KitchenQual"] = {'Po':1, 'Fa':2, 'TA':3, 'Gd':4, 'Ex':5}
+
+value_map["Functional"] = {'Sal':1, 'Sev':2, 'Maj2':3, 'Maj1':3, 'Mod':4, 'Min2':5, 'Min1':5, 'Typ':6}
+
+value_map["FireplaceQu"] = {'NA':1, 'Po':1, 'Fa':2, 'TA':3, 'Gd':4, 'Ex':5}
+
+value_map["GarageType"] = {'CarPort':1, 'NA':1,
+                           'Detchd':2,
+                           '2Types':3, 'Basment':3,
+                           'Attchd':4, 'BuiltIn':5}
+
+value_map["GarageFinish"] = {'NA':1, 'Unf':2, 'RFn':3, 'Fin':4}
+
+value_map["PavedDrive"] = {'N':1, 'P':2, 'Y':3}
+
+value_map["SaleType"] = {'COD':1, 'ConLD':1, 'ConLI':1, 'ConLw':1, 'Oth':1, 'WD':1,
+                        'CWD':2, 'VWD':2, 'Con':3, 'New':3}
+
+value_map["SaleCondition"] = {'AdjLand':1, 'Abnorml':2, 'Alloca':2, 'Family':2, 'Normal':3, 'Partial':4} 
+
+
+def fix_key(key):
+    #csv is wrong here
+    if key == 'Wd Shng':
+        key='WdShing'
+    if key == '2fmCon':
+        key='2FmCon'
+    if key == 'NAmes':
+        key='Names'
+    if key == 'Duplex':
+        key='Duplx'
+    if key == 'CmentBd':
+        key='CemntBd'
+    if key == 'C (all)':
+        key='C'
+    if key == 'Twnhs':
+        key='TwnhsI'
+    if key == 'Brk Cmn' or key =='BrkComm':
+        key='BrkCmn'
+    else:
+        key = key
+    return key
+
+miss_0 = ["PoolQC" , "MiscFeature", "Alley", "Fence", "FireplaceQu", "GarageQual", "GarageCond", "GarageFinish", "GarageYrBlt", "GarageType", "BsmtExposure", "BsmtCond", "BsmtQual", "BsmtFinType2", "BsmtFinType1", "MasVnrType"]
+miss_1=["MasVnrArea", "BsmtUnfSF", "TotalBsmtSF", "GarageCars", "BsmtFinSF2", "BsmtFinSF1", "GarageArea"]
+miss_2 = ['LotFrontage']
+def fix_miss(name):
+    if name in miss_0:
+        return 1
+    else:
+        return 0
+
+# def fix_LotFrontage(Full_map):
+#     a = np.zeros(25)
+#     for i in range(25):
+#         a[Full_map['Neighborhood'][i]-1] += 
+
+def add_future(features):
+    features["TotalHouse"] = features["TotalBsmtSF"] + features["1stFlrSF"] + features["2ndFlrSF"]   
+    features["TotalArea"] = features["TotalBsmtSF"] + features["1stFlrSF"] + features["2ndFlrSF"] + features["GarageArea"]
+    
+    features["TotalHouse_OverallQual"] = features["TotalHouse"] * features["OverallQual"]
+    features["GrLivArea_OverallQual"] = features["GrLivArea"] * features["OverallQual"]
+    features["my_MSZoning_TotalHouse"] = features["my_MSZoning"] * features["TotalHouse"]
+    features["my_MSZoning_OverallQual"] = features["my_MSZoning"] + features["OverallQual"]
+    features["my_MSZoning_YearBuilt"] = features["my_MSZoning"] + features["YearBuilt"]
+    features["my_Neighborhood_TotalHouse"] = features["my_Neighborhood"] * features["TotalHouse"]
+    features["my_Neighborhood_OverallQual"] = features["my_Neighborhood"] + features["OverallQual"]
+    features["my_Neighborhood_YearBuilt"] = features["my_Neighborhood"] + features["YearBuilt"]
+    features["BsmtFinSF1_OverallQual"] = features["BsmtFinSF1"] * features["OverallQual"]
+    
+    features["my_Functional_TotalHouse"] = features["my_Functional"] * features["TotalHouse"]
+    features["my_Functional_OverallQual"] = features["my_Functional"] + features["OverallQual"]
+    features["LotArea_OverallQual"] = features["LotArea"] * features["OverallQual"]
+    features["TotalHouse_LotArea"] = features["TotalHouse"] + features["LotArea"]
+    features["my_Condition1_TotalHouse"] = features["my_Condition1"] * features["TotalHouse"]
+    features["my_Condition1_OverallQual"] = features["my_Condition1"] + features["OverallQual"]
+    
+   
+    features["Bsmt"] = features["BsmtFinSF1"] + features["BsmtFinSF2"] + features["BsmtUnfSF"]
+    features["Rooms"] = features["FullBath"]+features["TotRmsAbvGrd"]
+    features["PorchArea"] = features["OpenPorchSF"]+features["EnclosedPorch"]+features["3SsnPorch"]+features["ScreenPorch"]
+    features["TotalPlace"] = features["TotalBsmtSF"] + features["1stFlrSF"] + features["2ndFlrSF"] + features["GarageArea"] + features["OpenPorchSF"]+features["EnclosedPorch"]+features["3SsnPorch"]+features["ScreenPorch"]
+    
+
+    features['YrBltAndRemod']=features['YearBuilt']+features['YearRemodAdd']
+    features['TotalSF']=features['TotalBsmtSF'] + features['1stFlrSF'] + features['2ndFlrSF']
+    features['Total_sqr_footage'] = (features['BsmtFinSF1'] + features['BsmtFinSF2'] +
+                                     features['1stFlrSF'] + features['2ndFlrSF'])
+    features['Total_Bathrooms'] = (features['FullBath'] + (0.5 * features['HalfBath']) +
+                                   features['BsmtFullBath'] + (0.5 * features['BsmtHalfBath']))
+
+    features['Total_porch_sf'] = (features['OpenPorchSF'] + features['3SsnPorch'] +
+                                  features['EnclosedPorch'] + features['ScreenPorch'] +
+                                  features['WoodDeckSF'])
+    # features['haspool'] = features['PoolArea'].apply(lambda x: 1 if x > 0 else 0)
+    # features['has2ndfloor'] = features['2ndFlrSF'].apply(lambda x: 1 if x > 0 else 0)
+    # features['hasgarage'] = features['GarageArea'].apply(lambda x: 1 if x > 0 else 0)
+    # features['hasbsmt'] = features['TotalBsmtSF'].apply(lambda x: 1 if x > 0 else 0)
+    # features['hasfireplace'] = features['Fireplaces'].apply(lambda x: 1 if x > 0 else 0)
+
+
+    return features
\ No newline at end of file

evaluation.py

@@ -2,6 +2,7 @@ import csv
 import math
 import numpy as np
 import dataloader
+import transformer
 
 def load_submission(path):
     csv_data = []
@@ -19,6 +20,8 @@ def eval_test(records_predict):
     return RMSE(records_real, records_predict)
 
 def RMSE(records_real,records_predict):
+    # records_real = np.log1p(records_real)
+    # records_predict = np.log1p(records_predict)
     records_real = dataloader.normlize(np.array(records_real),True)
     records_predict = dataloader.normlize(np.array(records_predict),True)
     if len(records_real) == len(records_predict):
@@ -28,7 +31,9 @@ def RMSE(records_real,records_predict):
         return None
 
 def main():
-    my_price = load_submission('./datasets/sample_submission.csv')
+    # my_price = load_submission('./datasets/sample_submission.csv')
+    my_price = load_submission('./result/0.03688_0.14435.csv')
+
     print(eval_test(my_price))
 if __name__ == '__main__':
     main()
\ No newline at end of file

ml.py

+from sklearn.linear_model import ElasticNet, Lasso,  BayesianRidge, LassoLarsIC
+from sklearn.ensemble import RandomForestRegressor,  GradientBoostingRegressor
+from sklearn.kernel_ridge import KernelRidge
+from sklearn.pipeline import make_pipeline
+from sklearn.preprocessing import RobustScaler
+from sklearn.base import BaseEstimator, TransformerMixin, RegressorMixin, clone
+from sklearn.model_selection import KFold, cross_val_score, train_test_split
+from sklearn.metrics import mean_squared_error
+from sklearn.model_selection import GridSearchCV
+# import xgboost as xgb
+# import lightgbm as lgb
+
+import numpy as np
+import torch
+import dataloader
+import evaluation
+import time
+import transformer
+
+dimension = 85
+train_desc,train_price,test_desc = dataloader.load_all(dimension)
+
+
+kr = GridSearchCV(KernelRidge(kernel='polynomial'),
+                  param_grid={"alpha": np.logspace(-3, 2, 6),
+                              "gamma": np.logspace(-2, 2, 5)})
+
+# print(np.logspace(-2, 2, 5))
+kr.fit(train_desc, train_price)
+y_kr = kr.predict(test_desc)
+for i in range(len(y_kr)):
+    y_kr[i] = dataloader.convert2price(y_kr[i])
+# print(y_kr.shape)
+print(dimension,evaluation.eval_test(y_kr))
+dataloader.write_csv(y_kr, './result/result.csv')
\ No newline at end of file

model.py

@@ -19,4 +19,4 @@ class Linear(nn.Module):
         x = self.relu(x)
         x = self.dropout(x)
         x = self.fc2(x)
-        return x
\ No newline at end of file
+        return x

train.py

@@ -5,7 +5,7 @@ import model
 import evaluation
 from torch import nn, optim
 import time
-
+import transformer
 
 #parameter
 LR = 0.0001
@@ -14,14 +14,14 @@ BATCHSIZE = 64
 CONTINUE = False
 use_gpu = True
 SAVE_FRE = 5
-
+Dimension = 120
 #load data
-train_desc,train_price,test_desc = dataloader.load_all()
+train_desc,train_price,test_desc = dataloader.load_all(Dimension)
 train_desc.tolist()
 train_price.tolist()
 
 #def network
-net = model.Linear(79,256,1)
+net = model.Linear(Dimension,256,1)
 print(net)
 
 if CONTINUE:
@@ -43,9 +43,13 @@ for epoch in range(EPOCHS):
     price_pres = []
     price_trues = []
 
-    dataloader.match_random(train_desc, train_price)
+    transformer.match_random(train_desc, train_price)
+    train_desc = np.array(train_desc)
+    train_price = np.array(train_price)
+    # train_desc = transformer.random_transform(train_desc, 0.02)
+    # train_price = transformer.random_transform(train_price, 0.02)
     for i in range(int(len(train_desc)/BATCHSIZE)):
-        desc = np.zeros((BATCHSIZE,79), dtype=np.float32)
+        desc = np.zeros((BATCHSIZE,Dimension), dtype=np.float32)
         price = np.zeros((BATCHSIZE,1), dtype=np.float32)
         for j in range(BATCHSIZE):   
             desc[j]=train_desc[i*BATCHSIZE+j:i*BATCHSIZE+j+1]
@@ -69,7 +73,7 @@ for epoch in range(EPOCHS):
     net.eval()
     price_pres = []
     for i in range(len(test_desc)):
-        desc = (test_desc[i]).reshape(1,79)
+        desc = (test_desc[i]).reshape(1,Dimension)
         desc = torch.from_numpy(desc).cuda()
         price_pre = net(desc)
         price_pres.append(dataloader.convert2price(price_pre.cpu().detach().numpy()[0][0]))

transformer.py

+import numpy as np
+import random
+
+def match_random(a,b):
+    state = np.random.get_state()
+    np.random.shuffle(a)
+    np.random.set_state(state)
+    np.random.shuffle(b)
+
+def random_transform(a,alpha):
+	return a*random.uniform(1-alpha,1+alpha)
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