Here we will present the setup for single machine to run time consuming machine learning experiments like feature selection using different machine learning models.
First we will create python program which runs single experiment.
We will use argparse library to be able to specify experiment parameters such as target variable, machine learning models to run and number of hyper parameter optimisation iterations, among others.
We will use logging module to log everything into one large text file.

The data for training will be read from the pickle files, as this is the fastest way to read the data.
The data will be created by an external program and then dataframe pickled to disk.

The structure of this python the following ,with example variables to include:

from sklearn.pipeline import Pipeline
from ci.fs import StandardScaler

import argparse
my_parser = argparse.ArgumentParser(description='run classification models on df with features')
my_parser.add_argument('-resample','-r',metavar='resample',type=str,help='resample 1Min 30S 5Min', dest="resample", default='30S')
my_parser.add_argument('-y',metavar='y',type=str,help='y = ybs ybb ycb ycs',dest="y", default='ycb')
my_parser.add_argument('-xs',metavar='xs',type=str,help='xs = all raw q3 q95 filename ',dest="xs", default='all')
my_parser.add_argument('-xsraw',metavar='xsraw',nargs='+',help='xsraw features',dest="xsraw", default=['m','amb','nbp','dpinsell','qbp','dpin','qsp','qimb','signimb','hml','vimb1','vimb5','cimb1','cimb5','vimb1000','cimb1000','vwap1','r'])


my_parser.add_argument('-models','--m', nargs='+', help='models to run = log lin xgb nn plog plin pn',dest="models", default=['log'])
my_parser.add_argument('-optiter',metavar='o',type=int,help='n_iter in CVrandsearch for all models',dest="optiter", default=100)

my_parser.add_argument('-test','--t',dest="test", default=False,action='store_true')
my_parser.add_argument('-addlog',dest="addlog", default=False,action='store_true')
my_parser.add_argument('-interact','--i',help='True is use interact features',dest="interact", default=False,action='store_true')
my_parser.add_argument('-scale',help='True to scale via pipeline ',dest="scale", default=False,action='store_true')


my_parser.add_argument('-gap',metavar='o',type=int,help='cv gap',dest="gap", default=100)
my_parser.add_argument('-max_train_size',metavar='o',type=int,help='cv max train size',dest="max_train_size", default=2000)

args = my_parser.parse_args()
logging.info(f"START - args={args}")
resample = args.resample
optiter=args.optiter
models=args.models
y = args.y
xs=args.xs
xsraw=args.xsraw
interact=args.interact
addlog=args.addlog
scale=args.scale

dffilename='dfbt'+resample+'f.pkl'#'dfbt30sf.pkl' dfbt30Sfandinter.pkl
pd.set_option("display.precision", 3)

df=pd.read_pickle(dffilename).ffill()
print(f"dfcols={list(df.columns)}")
#keep only float columns,  not time
#ipdb.set_trace()
df=df.select_dtypes(include=[np.float,np.int,np.int64,np.int32])
print(f"only floats={list(df.columns)}")

#linear regression ys
yrs=getfeaturenames('y',df.columns)
yrs=[yr for yr in yrs if df[yr].nunique()>10]
print(f"yrs={yrs}") 

xsraw=getfeaturenames('raw',df.columns,xsraw=xsraw)
xsq3=getfeaturenames('q3',df.columns,xsraw=xsraw)
xsq95=getfeaturenames('q95',df.columns,xsraw=xsraw)


xsall=getfeaturenames('all',df.columns)
xsinteract=getfeaturenames('interact',df.columns)

xs={'all':xsall,'raw':xsraw,'q3':xsq3,'q95':xsq95}[xs]

if addlog:
    _,lognames=df.addlog(cols=xs,inplace=True,retfnames=True)
    xs+=lognames

if interact:
    xs.extend(xsinteract)

if 'xgbc' in models or 'pxgbc' in models:#False:

    params={'scale_pos_weight': 100, 'n_estimators': 30, 'max_depth': 5, 'max_delta_step': 10, 'learning_rate': 0.1, 'colsample_bytree': 0.8, 'base_score': 0.1, 'alpha': 1}
    fixedparams=dict(objective ='binary:logistic')
    model=xgb.XGBClassifier(**fixedparams,**params)
    params = {
            'n_estimators':[10,100,200],
            'colsample_bytree': [ 0.8, 1.0],
            'max_depth': [5,10],
            'learning_rate':[0.01,0.1,1],
            'alpha':[1,10,100],
            'scale_pos_weight':[1,10,100],
            'base_score':[0.1,0.9],
            'max_delta_step':[0,1,10]
            }
    randcv = RandomizedSearchCV(model, param_distributions=params, n_iter=n_iter, scoring='f1', n_jobs=1, cv=cv, verbose=0, random_state=1).fit(dftrain[xs], dftrain[y])
    logging.info(f"rscv {model.__class__.__name__} fixedparams={fixedparams} bestscore={randcv.best_score_} bestparams={randcv.best_params_} \n{DF(randcv.cv_results_).sort_values(by='mean_test_score',ascending=False)[['mean_test_score' ,'std_test_score', 'params']]}")
    logging.debug(f"rscv \n{DF(randcv.cv_results_).sort_values(by='mean_test_score',ascending=False)}")
    xgbc=xgb.XGBClassifier(**fixedparams,**randcv.best_params_)


def getmodel(modelname):
    if modelname[0]=='p':
        return Pipeline([("pipe0",StandardScaler()),("pipe1",eval(modelname[1:]))])
    else:
        return eval(modelname)

for mstr in models:
    runselector(dftrain,y=y,xs=xs,model=getmodel(mstr),nansy='.fillna(0)',nansx=None,verbose=2,methods=['sfsb','sfsf','rfe','abscoef'],dftest=dftest,scoring='f1',eval_metric='f1',cv=cv)  #,

where we would use runselector function from the feature selection post.

We would then run this python file using windows bat files as following:

call python runexp.py -resample 30S -y ybb -xs raw  -models plog pxgbc
call python runexp.py -resample 30S -y ybs -xs raw  -models plog pxgbc
call python runexp.py -resample 30S -y ycb -xs raw  -models plog pxgbc

Feature selection in low signal-to-noise environments like finance.

In the following we will create a feature selection function which would work on XGBoost models as well as Tensorflow and simple sklearn models.
We will use univariate as well as other state of the art selection methods such as boruta,sequential feature elimination and shap values.

It’s important ot notice that in noisy environments different feature selection methods (and even same method, run twice) will not usually produce same sets of features.
Thus we will measure weighted tau rank correlation between sets of features produced by different methods and same methods , but on train and test sets.
We will use weighted and not simple tau correlation to emphasize top ranked features.

Feature selection is usually the most time-consuming step in machine learning applications, thus we will be logging the progress to file using python logging module.

Another point to mention is that it’s useful to add non-informative “noise” features into the set of actual features, and then look into rank position of these noise features to measure the performance of the the feature selection algorithm.

For univariate feature selection we would recommend just using distance correlation (to measure non-linear dependence effects) and pearson correlation (for linear dependence) , as other methods, such as Fisher F-test, Chi2 and tau and spearman correlations would give similar results.

We will use the following python packages:

 pip install xgboost
 pip install logging
 pip install mlextend
 pip install eli5
 pip install pyentrp
 
 

The function signature will be the following:

 def runselector(df,y,xs,model,nansy,nansx,methods=None,scoring=None,eval_metric=None,verbose=0,cv=None,eliniter=5,dftest=None):
 

where df is a dataframe (train set)
y is column name for y variable (we predict y ~ xs )
xs is list of feature column names.
nansy = NaN’s substitution rule for y
nansx NaN’s substitution rule for features
methods = list of feature selection methods
scoring = scoring function (i.e. f1)
cv = cross validation iterator
eliniter = number of iterations for eli method
dftest = test dataset (optional)

We will also make this function to work with sklearn’s pipelines, useful when features need scaling, and to avoid information leak from scaling whole dataset.

To run univariate feature selection inside feature selector we will use help function fs1().

def fs1(res,y,xs=None):
    if xs is None:
        xs=res.columns
    xs=list(xs)
    xs=list(set(xs)-set([y]))
    df=res[xs+[y]]
    dcors={}
    pearsons={}
    fctests={}
    frtests={}
    chi2abs={}
    mir={}
    mic={}
    kendalls={}
    for col in xs:
        dfdropna=df[[col,y]].replace([np.inf, -np.inf], np.nan).dropna()
        if dfdropna.shape==(0,2):
            continue
        try:
            fctests[col]=f_classif(dfdropna[[col]],dfdropna[y])[0]
        except:
            pass

        dcors[col]=dcor(dfdropna[col],dfdropna[y])
        pearsons[col]=dfdropna[[col,y]].corr().iloc[1,0]

    corrs=DF(pearsons,index=['pearsonabs']).abs().T.join(DF(dcors,index=['dcor']).T).join(DF(fctests,index=['fc']).T)
    res=pd.concat([corrs,corrs.rank(pct=True).add_prefix('rk.')],axis=1).sort_values(by='rk.dcor',ascending=False) 
    logging.info(f"fs1 rk.dcor index {list(res['rk.dcor'].index)}")
    return res

Example of usage on dummy classification problem with target variable y and features x1,x2,x3 and additional feaature q.x3 (quantile of x3 noise feature) :

from sklearn.ensemble import RandomForestClassifier
import xgboost as xgb
import numpy as np
from dcor import distance_correlation as dcor
from eli5.sklearn import PermutationImportance
from mlxtend.feature_selection import SequentialFeatureSelector
from pyentrp.entropy import permutation_entropy as pentropy
import shap
from scipy.stats import weightedtau as wtau
npa=np.array
DF=pd.DataFrame
import logging

modelxgb=xgb.XGBClassifier(objective ='binary:logistic', colsample_bytree = 1, learning_rate = 1,max_depth = 10, alpha = 1, n_estimators = 5)
x1=npa([-1,-2,-3,-4,-5,6,7,8,9,10,11,12,13,14,15])
np.random.shuffle(x1)
x2=x1+np.random.randn(len(x1))
x3=np.random.randn(len(x1))
y=(x1>0).astype(int)
xs=['x1','x2','x3','q.x3']
dfunittest=DF({'x1':x1,'x2':x2,'x3':x3,'q.x3':pd.qcut(x3,3,labels=False),'y':y})
display(fs1(dfunittest,'y',xs))
runselector(dfunittest.iloc[:10],y='y',xs=xs,model=modelxgb,nansy='.fillna(0)',nansx=None,verbose=10,methods=['sfsb','sfsf','eli','shap'],dftest=dfunittest.iloc[-10:],scoring='f1',eval_metric='auc',cv=2)

output:

Full code of the feature selection and helper functions:

from sklearn.ensemble import RandomForestClassifier
import xgboost as xgb
import numpy as np
from dcor import distance_correlation as dcor
from eli5.sklearn import PermutationImportance
from mlxtend.feature_selection import SequentialFeatureSelector
from pyentrp.entropy import permutation_entropy as pentropy
import shap
from scipy.stats import weightedtau as wtau
npa=np.array
DF=pd.DataFrame
import logging


def calccorr(df,method='dcor',**kwargs):
    #ipdb.set_trace()
    if method.replace('abs','') in ['pearson','kendall','spearman']:
        dfres=df.corr(method=method.replace('abs','')).rename_axis(method)
        if 'abs' in method:
            return dfres.abs()
        else:
            return dfres
    cols=df.columns
    resd=DF(np.zeros((len(cols),len(cols))),index=cols,columns=cols)
    for icol1 in range(len(cols)):
        for icol2 in range(len(cols)):
            if icol1!=icol2:
                dfdropna=df.iloc[:,[icol1,icol2]].dropna()
                res=eval(method)(dfdropna.iloc[:,0],dfdropna.iloc[:,1],**kwargs)
                if method in ['wtau']:
                    res=res[0]
                if method in ['np.corrcoef']:
                    res=res[1,0]
                resd.iloc[icol1,icol2]=res
            else: #calc pentropy
                #ipdb.set_trace()
                resd.iloc[icol1,icol2]=pentropy(df.iloc[:,[icol1]].dropna().values.flatten(),order=3,delay=1,normalize=True)                
    return resd.rename_axis(method)

pd.core.frame.DataFrame.calccorr=calccorr 


def myround2(ts):
        try:
            return int(np.round(float(ts),2)*100)
        except: pass
        try:
            return (np.round(ts.astype(float),2)*100).astype(int)
        except: pass
        try:
            return {k:(np.round(v,2)*100).astype(int) for k,v in ts.items() }
        except Exception as e:
            pass
        return ts

pd.core.frame.DataFrame.myround2=lambda df:df.applymap(myround2)
pd.core.series.Series.myround2=lambda df:df.apply(myround2)

def fs1(res,y,xs=None):
    if xs is None:
        xs=res.columns
    xs=list(xs)
    xs=list(set(xs)-set([y]))
    df=res[xs+[y]]
    dcors={}
    pearsons={}
    fctests={}
    frtests={}
    chi2abs={}
    mir={}
    mic={}
    kendalls={}
    for col in xs:
        dfdropna=df[[col,y]].replace([np.inf, -np.inf], np.nan).dropna()
        if dfdropna.shape==(0,2):
            continue
        try:
            fctests[col]=f_classif(dfdropna[[col]],dfdropna[y])[0]
        except:
            pass

        dcors[col]=dcor(dfdropna[col],dfdropna[y])
        pearsons[col]=dfdropna[[col,y]].corr().iloc[1,0]

    corrs=DF(pearsons,index=['pearsonabs']).abs().T.join(DF(dcors,index=['dcor']).T).join(DF(fctests,index=['fc']).T)
    res=pd.concat([corrs,corrs.rank(pct=True).add_prefix('rk.')],axis=1).sort_values(by='rk.dcor',ascending=False) 
    logging.info(f"fs1 rk.dcor index {list(res['rk.dcor'].index)}")
    return res

def runselector(df,y,xs,model,nansy,nansx,methods=None,scoring=None,eval_metric=None,verbose=0,cv=None,eliniter=5,dftest=None):
    try:    
        if 'xgb' in model.named_steps['pipe1'].__class__.__name__.lower():
            if eval_metric=='f1':
                eval_metric=minusf1
    except:
        if 'xgb' in 'xgb' in model.__class__.__name__.lower():
            if eval_metric=='f1':
                eval_metric=minusf1
        
    if methods is None:
        methods=['boruta','rfe','sfsb','sfsf','shap','eli']
    
    xs=list(xs)
    if scoring is None:
        if df[y].nunique()<10:
            scoring='balanced_accuracy'
        else:
            scoring='r2'
        print('runselector scoring is None.using {}'.format(scoring))
    
    ytrain=eval('df[y]'+nansy)
    xtrain=eval('df[xs]'+nansx) if nansx is not None else df[xs]
    try:
        inferfreq=pd.infer_freq(df.index)
    except:
        inferfreq=None

    logging.info(f"runselector START df.index.min,max={df.index.min(),df.index.max()} dftest.minmax={dftest.index.min(),dftest.index.max() if dftest is not None else 'None'}  inferfreq={inferfreq} meansecsdiff= {float(np.diff(npa(df.index)).mean())/1e9}secs scoring={scoring} \n modelclass={model.__class__.__name__} modeldict={model.__dict__} xs={xs} \n {df.describe()}")
    
    try:
        model=eval(model)
    except Exception as e:
        print(e)
           
    fs1df=fs1(df,y,xs)
    res=fs1df[fs1df.columns[fs1df.columns.str.contains('rk\\.')]]
    
    if 'boruta' in methods:

        boruta_selector=BorutaPy(model).fit(xtrain,ytrain)#, n_estimators = 10, random_state = 0)
  #      boruta_selector=BorutaPy(model).fit(xtrain.values,ytrain.values)#, n_estimators = 10, random_state = 0)
        boruta=DF({'boruta':boruta_selector.ranking_,'xs':xs}).set_index('xs').rank(ascending=False,pct=True).sort_values(by='boruta',ascending=False)
        logging.info(f'boruta: {boruta.round(2)*100}')
        res=res.join(boruta)
    #boruta_selector = selectormodel
    
    if 'abscoef' in methods:
        try:
#            ipdb.set_trace()
            modelcoef=clone(model)
            modelcoef.fit(xtrain, ytrain)
            rfeselectorranking=np.abs(modelcoef.coef_[0])*xtrain.std() #multiply by feature stddev, ocnditional that feture is centered at 0
            abscoef=DF({'abscoef':rfeselectorranking,'xs':xs}).set_index('xs').rank(ascending=False,pct=True)
            res=res.join(abscoef)
            abscoeflog=DF({'abscoefbystd':rfeselectorranking,'coef':modelcoef.coef_[0],'std':xtrain.std(),'xs':xs}).set_index('xs').sort_values(by='abscoefbystd',ascending=False)
            logging.info(f'abscoef:\n {abscoeflog}')
        except Exception as e:
            print(f"abscoef:{e}")

    if 'rfe' in methods:
        try:
            rfeselector = RFE(model, 1, step=1).fit(xtrain, ytrain)
            rfeselectorranking=rfeselector.ranking_
            rfe=DF({'rfe':rfeselectorranking,'xs':xs}).set_index('xs').rank(ascending=False,pct=True)
            res=res.join(rfe)
        except Exception as e:
            print(f"rfe:{e}")
    if 'sfsf' in methods:
        sfsf = SequentialFeatureSelector(model,k_features=len(xs), forward=True, floating=False,  verbose=0,  scoring=scoring,  cv=cv).fit(xtrain, ytrain,custom_feature_names=xs)
        sfsF=DF(np.unique(DF(sfsf.get_metric_dict()).T['feature_names'].sum(), return_counts=True)).T.set_index(0).rank(pct=True).sort_values(by=1,ascending=False).rename(columns={1:'sfsF'})
        if verbose>1: 
            display(DF(sfsf.get_metric_dict()).T[['avg_score','cv_scores','std_dev','feature_names']])
        res=res.join(sfsF)
        logging.info(f'sfsf:\n {sfsF.round(2)*100}')
    if 'sfsb' in methods:
        sfsb = SequentialFeatureSelector(model,k_features=1, forward=False, floating=False,  verbose=0,  scoring=scoring,  cv=cv).fit(xtrain, ytrain,custom_feature_names=xs)
        sfsB=DF(np.unique(DF(sfsb.get_metric_dict()).T['feature_names'].sum(), return_counts=True)).T.set_index(0).rank(pct=True).sort_values(by=1,ascending=False).rename(columns={1:'sfsB'})

        if verbose>1:
            sfsbd=DF(sfsb.get_metric_dict()).T[['avg_score','cv_scores','std_dev','feature_names']]
            if dftest is not None:
                sfsbd['dftest']=''
                
                for i,row in sfsbd.iterrows():
                    if 'Pipe' in model.__class__.__name__:
                        model.fit(df[list(row['feature_names'])],df[y],pipe1__eval_metric=eval_metric,pipe1__eval_set=[(dftest[list(row['feature_names'])], dftest[y])],pipe1__verbose=0)
                        sfsbd.at[i,'dftest'] = model.named_steps['pipe1'].evals_result()['validation_0']
                    else:
                        model.fit(df[list(row['feature_names'])],df[y],eval_metric=eval_metric,eval_set=[(dftest[list(row['feature_names'])], dftest[y])],verbose=0)
                        sfsbd.at[i,'dftest'] = model.evals_result()['validation_0']
                    
                    print(list(row['feature_names']),eval_metric,sfsbd.at[i,'dftest'])
            logging.info(f"sfsbd=\n{sfsbd.myround2()}")
            display(sfsbd.round(3))
        logging.info(f'sfsb:\n {sfsB.round(2)*100}')
        res=res.join(sfsB)
        
    model.fit(xtrain,ytrain)
    
    if 'eli' in methods:
        if cv is None:
            cv='prefit'
        permuter = PermutationImportance(model, scoring=None, cv=cv, n_iter=eliniter, random_state=42)#instantiate permuter object #'balanced_accuracy'  'prefit'
        elidf=DF({'eli':permuter.fit(xtrain.values,ytrain.values).feature_importances_,'xs':xs}).set_index('xs').rank(ascending=True,pct=True).sort_values(by='eli',ascending=False)
        logging.info(f'eli: {elidf.round(2)*100}')
        res=res.join(elidf)
            
    if 'shap' in methods:
        if 'Pipe' in model.__class__.__name__:
            if 'xgb' in model.named_steps['pipe1'].__class__.__name__.lower() and model.named_steps['pipe1'].get_params()['booster'] in ('gbtree',None):
                explainer=shap.TreeExplainer(model.named_steps['pipe1'],model_output='raw')
            elif 'NN' in model.named_steps['pipe1'].__class__.__name__:
                explainer=shap.DeepExplainer(model.named_steps['pipe1'].model.model, data=xtrain.values)#, session=None, learning_phase_flags=None)
            elif 'Logistic' in model.named_steps['pipe1'].__class__.__name__:
                explainer=shap.KernelExplainer(model.named_steps['pipe1'].predict_proba, data=xtrain.values, link='logit',l1_reg='aic')
            else:
                raise ValueError(f"shap for model {model.named_steps['pipe1'].__class__.__name__} {model.named_steps['pipe1'].get_params()} not imlemented in fs.py")
        else:
            if 'xgb' in model.__class__.__name__.lower() and model.get_params()['booster'] in ('gbtree',None):
                explainer=shap.TreeExplainer(model,model_output='raw')
            elif 'NN' in model.__class__.__name__:
                explainer=shap.DeepExplainer(model.model.model, data=xtrain.values)#, session=None, learning_phase_flags=None)
            elif 'Logistic' in model.__class__.__name__:
                explainer=shap.KernelExplainer(model.predict_proba, data=xtrain.values, link='logit',l1_reg='aic')
            else:
                raise ValueError(f"shap for model {model.__class__.__name__} and params={model.get_params()} not imlemented in fs.py")

        try:

            shap_values = explainer.shap_values(xtrain.values)#, tree_limit=5)
            concat=np.concatenate(shap_values) if type(shap_values)==type([]) else shap_values
            shap_abs = np.abs(concat)
            global_importances = np.nanmean(shap_abs, axis=0)
            indices = np.argsort(global_importances)[::-1]
            features_ranked = []
            for f in range(df[xs].shape[1]):
                features_ranked.append(xs[indices[f]])
            shapdf=DF({'shap':global_importances},index=xs).rank(ascending=True,pct=True)
            res=res.join(shapdf)
            if verbose>2:
                shap.summary_plot(shap_values, df[xs], plot_type="bar",class_names=model.classes_)
                shap.initjs()
                try:
                    for i in range(len(explainer.expected_value)):
                        display(shap.force_plot(explainer.expected_value[i], shap_values[i],df[xs]))
                except Exception as e:
                    print(f"forceplot exception:{e}")
        except Exception as e:
            logging.warning(f"shaps exception = {e}")            
            
    res['mean']=res.mean(axis=1)
    res=res.sort_values(by='mean',ascending=False)
    
    if verbose>1:
        logging.info(f"res.corr.mean=\n{100*res.corr().round(2)}")
        display(100*res.corr().round(2))
        print(f"meancorr=\n{res.corr().mean().round(2)*100}")
        logging.info(f"meancorr=\n{res.corr().mean().myround2()}")
        
    if verbose>1:
        res1=res.copy()
        res1['pfname']=res1.index.str.split('.').str[-1]
        res1=res1.groupby('pfname').mean().sort_values(by='mean',ascending=False)#.set_index('pfname')
        print("pure features mean rank")
        display(res1)
        logging.info(f"pure fs mean rank\n {res1.myround2()}")
        print("pure features wtau")
        try:
            display(100*res1.calccorr(method='wtau').round(2))
            logging.info(f"pure features wtau \n{res1.calccorr(method='wtau').myround2()}")
        except Exception as e:
            print(f"wtau calcorr exception{str(e)}")
     
    logging.info(f"runselector complete res=\n{res.round(2)*100} {list(res.index)}")
    return res

To run the feature selection on the scaled features one can use the same function and pipe as following:

model=Pipeline([("pipe0",StandardScaler()),("pipe1",xgb.XGBClassifier(objective ='binary:logistic', colsample_bytree = 1, learning_rate = 1,max_depth = 10, alpha = 1, n_estimators = 5))])

In many time series machine learning problems the with large number of features the raw data might contain

– abnormal / extreme points
– discontinuities
– stale data

To help with determining quickly abnormal or extreme points we can use z-transform of the time series.
To dtermine if time series contain discountinuities we can calculate how much removing one point changes sum of first difference of the time series.
and lastly to determine stale/predictable data we can use permutation-entropy implemented in python package pyentrp.

The code to run all the 3 tests at once is present below:

from scipy.stats import chi2 
from pyentrp.entropy import permutation_entropy as pentropy

def smoothcoef(y):
    y=npa(y).flatten()
    ymax=max(np.abs(y).max(),0.0000001)
    tv=np.zeros(len(y))
    for i in range(len(y)-1):
        try:
            tv[i]=np.abs(np.diff(np.delete(y,i),1)).sum()
        except:
            ipdb.set_trace()
    tv=np.delete(tv,[0,len(tv)-1])
    tv=np.abs(np.diff(tv,1))
    return tv.max()/ymax

def abnormality(ts,thresh,retpoints=False,plot=False): #high values means abnormal
        avg=ts.mean()
        var=ts.var()
        nans=ts.isnull().sum()/len(ts)
        abnormal=(ts-avg)**2/var >chi2.interval(1-thresh, 1)[1]
        if plot:
            if (plot=='abnormal' and abnormal.any()) or plot=='all':                
                plt.figure(figsize = (4, 4))
                plt.clf()
                plt.scatter(ts.index,ts.values,c=abnormal,cmap='bwr',marker='.') 
                plt.show()
        res={}
        if retpoints:
            res['abnpoints']=ts[abnormal]
        return {**res,'abnormal':float(abnormal.any()),'nans':nans,'nunique':1-ts.nunique()/len(ts),'smooth':smoothcoef(ts.dropna()),'pentr':1-pentropy(ts.dropna(),normalize=True)}

Example usage is shown in the following code:

DF=pd.DataFrame
df=DF({'x':np.linspace(1,10,30)})
df['y']=np.sin(df['x'])
df
df['y'].iloc[4]=6
df[['y']].plot()
abnormality(df['y'],0.001,retpoints=False,plot='abnormal')

the result is show on the following figure:

Interpretation of the results is the following:

Higher the number => more probability there is abnormality in the time series.
When plot parameter is specified graph will be shown with abnormal points in red.

Sometimes it is preferable to use simple machine learning algorithms such as logistic regression due to speed and explainability.
But usually these simple algorithms do not incorporate interactions of the features (in contrary to , say, neural networks, where sum/difference of the features is incorporated automatically, as each neuron would sum up incoming connections, and using log transform also can add product/division of the features).

Thus here we present simple way to add feature interactions into machine learning pipeline:

def addinteract(df,cols=None,inplace=False,ops=None,retfnames=False):
    df = df if inplace else df.copy()
    fnames=[]
    if cols is None:
        cols=df.columns
    if ops is None:
        ops=['sum','sub','prd']
    def sum(a,b):
        return a+b
    def sub(a,b):
        return a-b
    def prod(a,b):
        return a*b
    for i in range(len(cols)):
        for j in range(i+1,len(cols)):
            for op in ops:
                try:
                    fname=op+'('+cols[i]+','+cols[j]+')'
                    df[fname]=eval(op)(df[cols[i]],df[cols[j]])
                    fnames.append(fname)
                except Exception as e:
                    print(e)
    if retfnames:
        return df,fnames
    return df
pd.core.frame.DataFrame.addinteract=addinteract

We can use it in the following way:

xint=np.random.randint(0,10,20)
DF=pd.DataFrame
DF({'x':xint,'y':xint-1,'z':xint+2}).addinteract()

which results in: