models/distance.py

"""Classes of distance measure for model type A
"""
# Author: Yinchen Wu <[email protected]>

import numpy as np
from arch import arch_model
import math


class Euclidean:
    """ The function class for Lp euclidean norm
    ----------
    Power : int, optional (default=1)
        The power of the lp norm. For power = k, the measure is calculagted by |x - y|_k
    neighborhood : int, optional (default=max (100, 10*window size))
        The length of neighborhood to derivete the normalizing constant D which is based on
        the difference of maximum and minimum in the neighborhood minus window. 
    window: int, optional (default = length of input data)
        The length of the subsequence to be compaired
    Attributes
    ----------
    decision_scores_ : numpy array of shape (n_samples,)
        The outlier scores of the training data.
        The higher, the more abnormal. Outliers tend to have higher
        scores. This value is available once the detector is
        fitted.
    detector: Object classifier
        the anomaly detector that is used
    """
    def __init__(self, power = 1, neighborhood = 100, window = 20, norm = False):
        self.power = power
        self.window = window
        self.neighborhood = neighborhood
        self.detector = None
        self.decision_scores_  = []
        self.norm = norm
        self.X_train = 2
    def measure(self, X, Y, index):
        """Derive the decision score based on the given distance measure 
        Parameters
        ----------
        X : numpy array of shape (n_samples, )
            The real input samples subsequence.
        Y : numpy array of shape (n_samples, )
            The estimated input samples subsequence.
        Index : int
        the index of the starting point in the subsequence
        Returns
        -------
        score : float
            dissimiarity score between the two subsquence
        """
        X_train = self.X_train
        X_train = self.detector.X_train_
        power = self.power
        
        window = self.window
        neighborhood = self.neighborhood
        norm = self.norm
        data = X_train
        if norm == False:
            if X.shape[0] == 0:
                score = 0
            else:
                score = np.linalg.norm(X-Y, power)/(X.shape[0])
            self.decision_scores_.append((index, score))
            return score
        elif type(X_train) == int:
            print('Error! Detector is not fed to the object and X_train is not known')
        elif neighborhood != 'all':
            neighbor = int(self.neighborhood/2)

            if index + neighbor < self.n_train_ and index - neighbor > 0: 
                region = np.concatenate((data[index - neighbor: index], data[index + window: index + neighbor] ))
                D = np.max(region) - np.min(region)
            elif index + neighbor >= self.n_train_ and index + window < self.n_train_:
                region = np.concatenate((data[self.n_train_ - neighborhood: index], data[index + window: self.n_train_] ))
                D =  np.max(region) - np.min(region)   
            elif index + window >= self.n_train_:
                region = data[self.n_train_ - neighborhood: index]
                D = np.max(region) - np.min(region) 
            else:
                region = np.concatenate((data[0: index], data[index + window: index + neighborhood] ))
                D = np.max(region) - np.min(region) 
            
            score = np.linalg.norm(X-Y, power)/D/(X.shape[0]**power)
            self.decision_scores_.append((index, score))
            return score
    def set_param(self):
        if self.detector != None:
            self.window = self.detector.window
            self.neighborhood = self.detector.neighborhood
            self.n_train_ = self.detector.n_train_
            self.X_train = self.detector.X_train_
        else:
            print('Error! Detector is not fed to the object and X_train is not known')
        return self
                

class Mahalanobis:
    """ The function class for Mahalanobis measure
    ----------
    Probability : boolean, optional (default=False)
        Whether to derive the anomoly score by the probability that such point occurs
    neighborhood : int, optional (default=max (100, 10*window size))
        The length of neighborhood to derivete the normalizing constant D which is based on
        the difference of maximum and minimum in the neighborhood minus window. 
    Attributes
    ----------
    decision_scores_ : numpy array of shape (n_samples,)
        The outlier scores of the training data.
        The higher, the more abnormal. Outliers tend to have higher
        scores. This value is available once the detector is
        fitted.
    detector: Object classifier
        the anomaly detector that is used
    """
    def __init__(self, probability = False):
        self.probability = probability
        self.detector = None
        self.decision_scores_  = []
        self.mu = 0
        
    def set_param(self):
        '''update the parameters with the detector that is used 
        '''

        self.n_initial_ = self.detector.n_initial_
        self.estimation = self.detector.estimation
        self.X_train = self.detector.X_train_
        self.window = self.detector.window
        window = self.window
        resid = self.X_train - self.estimation
        number = max(100, self.window)
        self.residual = np.zeros((window, number))
        for i in range(number):
            self.residual[:, i] = resid[self.n_initial_+i:self.n_initial_+i+window]
        self.mu = np.zeros(number)
        self.cov = np.cov(self.residual, rowvar=1)
        if self.window == 1:
            self.cov = (np.sum(np.square(self.residual))/(number - 1))**0.5
        return self
    def norm_pdf_multivariate(self, x):
        '''multivarite normal density function
        '''
        try:
            mu = self.mu
        except:
            mu = np.zeros(x.shape[0])
        sigma = self.cov
        size = x.shape[0]
        if size == len(mu) and (size, size) == sigma.shape:
            det = np.linalg.det(sigma)
            if det == 0:
                raise NameError("The covariance matrix can't be singular")

            norm_const = 1.0/ ( math.pow((2*math.pi),float(size)/2) * math.pow(det,1.0/2) )
            x_mu = np.matrix(x - mu)
            inv = np.linalg.inv(sigma)        
            result = math.pow(math.e, -0.5 * (x_mu * inv * x_mu.T))
            return norm_const * result
        else:
            raise NameError("The dimensions of the input don't match")
    def normpdf(self, x):
        '''univariate normal
        '''
        mean = 0
        sd = np.asscalar(self.cov)
        var = float(sd)**2
        denom = (2*math.pi*var)**.5
        num = math.exp(-(float(x)-float(mean))**2/(2*var))
        return num/denom 

    def measure(self, X, Y, index):
        """Derive the decision score based on the given distance measure 
        Parameters
        ----------
        X : numpy array of shape (n_samples, )
            The real input samples subsequence.
        Y : numpy array of shape (n_samples, )
            The estimated input samples subsequence.
        Index : int
        the index of the starting point in the subsequence
        Returns
        -------
        score : float
            dissimiarity score between the two subsquence
        """
        mu = np.zeros(self.detector.window)
        cov = self.cov
        if self.probability == False:

            if X.shape[0] == mu.shape[0]:
                score = np.matmul(np.matmul((X-Y-mu).T, cov), (X-Y-mu))/(X.shape[0])
                self.decision_scores_.append((index, score))
                return score
            else:
                return (X-Y).T.dot(X-Y)

        else:
            if len(X) > 1:
                prob = self.norm_pdf_multivariate(X-Y)
            elif len(X) == 1: 
                X = np.asscalar(X)
                Y = np.asscalar(Y)
                prob = self.normpdf(X-Y)
            else:
                prob = 1
            score = 1 - prob
            score = max(score, 0)
            self.decision_scores_.append((index, score))
            return score


class Garch:
    """ The function class for garch measure
    ----------
    p, q : int, optional (default=1, 1)
        The order of the garch model to be fitted on the residual
    mean : string, optional (default='zero' )
        The forecast conditional mean. 
    vol: string, optional (default = 'garch')
        he forecast conditional variance.
    Attributes
    ----------
    decision_scores_ : numpy array of shape (n_samples,)
        The outlier scores of the training data.
        The higher, the more abnormal. Outliers tend to have higher
        scores. This value is available once the detector is
        fitted.
    detector: Object classifier
        the anomaly detector that is used
    """
    def __init__(self, p = 1, q = 1, mean = 'zero', vol = 'garch'):
        self.p = p
        self.q = q
        self.vol = vol
        self.mean = mean
        self.decision_scores_  = []
        
    def set_param(self):
        '''update the parameters with the detector that is used 
        '''
        q = self.q
        p=self.p
        mean = self.mean
        vol = self.vol
        if self.detector != None:
            self.n_initial_ = self.detector.n_initial_
            self.estimation = self.detector.estimation
            self.X_train = self.detector.X_train_
            self.window = self.detector.window
            resid = 10 * (self.X_train - self.estimation)
            model = arch_model(resid, mean=mean, vol=vol, p=p, q=q)
            model_fit = model.fit(disp='off')
            self.votility = model_fit.conditional_volatility/10
        else:
            print('Error! Detector not fed to the measure')
        return self

    def measure(self, X, Y, index):
        """Derive the decision score based on the given distance measure 
        Parameters
        ----------
        X : numpy array of shape (n_samples, )
            The real input samples subsequence.
        Y : numpy array of shape (n_samples, )
            The estimated input samples subsequence.
        Index : int
        the index of the starting point in the subsequence
        Returns
        -------
        score : float
            dissimiarity score between the two subsquences
        """
        X = np.array(X)
        Y = np.array(Y)
        length = len(X)
        score = 0
        if length != 0:
            for i in range(length):
                sigma = self.votility[index + i]
                if sigma != 0:
                    score += abs(X[i]-Y[i])/sigma
                    
            score = score/length       
        return score


class SSA_DISTANCE:
    """ The function class for SSA measure
    good for contextual anomolies
    ----------
    method : string, optional (default='linear' )
        The method to fit the line and derives the SSA score
    e: float, optional (default = 1)
        The upper bound to start new line search for linear method
    Attributes
    ----------
    decision_scores_ : numpy array of shape (n_samples,)
        The outlier scores of the training data.
        The higher, the more abnormal. Outliers tend to have higher
        scores. This value is available once the detector is
        fitted.
    detector: Object classifier
        the anomaly detector that is used
    """
    def __init__(self, method ='linear', e = 1):
        self.method = method
        self.decision_scores_  = []
        self.e = e
    def Linearization(self, X2):
        """Obtain the linearized curve.
        Parameters
        ----------
        X2 : numpy array of shape (n, )
            the time series curve to be fitted
        e: float, integer, or numpy array 
        weights to obtain the 
        Returns
        -------
        fit: parameters for the fitted linear curve
        """
        e = self.e
        i = 0
        fit = {}
        fit['index'] = []
        fit['rep'] = []
        while i < len(X2):
            fit['index'].append(i)
            try:
                fit['Y'+str(i)]= X2[i]
            except:
                print(X2.shape, X2)
            fit['rep'].append(np.array([i, X2[i]]))
            if i+1 >= len(X2):
                    break
            k = X2[i+1]-X2[i]
            b = -i*(X2[i+1]-X2[i])+X2[i]
            fit['reg' +str(i)]= np.array([k, b])
            i += 2
            if i >= len(X2):
                break
            d = np.abs(X2[i]- (k * i+b))
            while d < e:
                i +=1 
                if i >= len(X2):
                    break
                d = np.abs(X2[i]- (k * i+b)) 
        return fit   
    def set_param(self):
        '''update the parameters with the detector that is used. 
        Since the SSA measure doens't need the attributes of detector
        or characteristics of X_train, the process is omitted. 
        '''

        return self

    def measure(self, X2, X3, start_index):
        """Obtain the SSA similarity score.
        Parameters
        ----------
        X2 : numpy array of shape (n, )
            the reference timeseries
        X3 : numpy array of shape (n, )
            the tested timeseries
        e: float, integer, or numpy array 
        weights to obtain the 
        Returns
        -------
        score: float, the higher the more dissimilar are the two curves 
        """       
        #linearization of data X2 and X3
        X2 = np.array(X2)
        X3 = np.array(X3)

        fit = self.Linearization(X2)
        fit2 = self.Linearization(X3)
    
        #line alinement 
        Index = []
        test_list = fit['index'] + fit2['index']
        [Index.append(x) for x in test_list if x not in Index]
        Y = 0
    
        #Similarity Computation
        for i in Index:
            if i in fit['index'] and i in fit2['index']:
                Y += abs(fit['Y'+str(i)]-fit2['Y'+str(i)])

            elif i in fit['index']:
                J = np.max(np.where(np.array(fit2['index']) < i ))
                index = fit2['index'][J]
                k = fit2['reg'+str(index)][0]
                b = fit2['reg'+str(index)][1]
                value = abs(k * i + b - fit['Y'+str(i)])
                Y += value
            elif i in fit2['index']:
                J = np.max(np.where(np.array(fit['index']) < i ))
                index = fit['index'][J]
                k = fit['reg'+str(index)][0]
                b = fit['reg'+str(index)][1]
                value = abs(k * i + b - fit2['Y'+str(i)])
                Y += value
        if len(Index) != 0: 
            score = Y/len(Index)
        else:
            score = 0
        self.decision_scores_.append((start_index, score))
        if len(X2) == 1:
            print('Error! SSA measure doesn\'t apply to singleton' )
        else:
            return score  


class Fourier:
    """ The function class for Fourier measure
    good for contextual anomolies
    ----------
    power: int, optional (default = 2)
        Lp norm for dissimiarlity measure considered
    Attributes
    ----------
    decision_scores_ : numpy array of shape (n_samples,)
        The outlier scores of the training data.
        The higher, the more abnormal. Outliers tend to have higher
        scores. This value is available once the detector is
        fitted.
    detector: Object classifier
        the anomaly detector that is used
    """
    def __init__(self, power = 2):
        self.decision_scores_  = []
        self.power = power
    def set_param(self):
        '''update the parameters with the detector that is used 
        since the FFT measure doens't need the attributes of detector
        or characteristics of X_train, the process is omitted. 
        '''

        return self

    def measure(self, X2, X3, start_index):
        """Obtain the SSA similarity score.
        Parameters
        ----------
        X2 : numpy array of shape (n, )
            the reference timeseries
        X3 : numpy array of shape (n, )
            the tested timeseries
        index: int, 
        current index for the subseqeuence that is being measured 
        Returns
        -------
        score: float, the higher the more dissimilar are the two curves 
        """       
 
        power = self.power
        X2 = np.array(X2)
        X3 = np.array(X3)
        if len(X2) == 0:
            score = 0
        else:
            X2 = np.fft.fft(X2)
            X3 = np.fft.fft(X3)
            score = np.linalg.norm(X2 - X3, ord = power)/len(X3)
        self.decision_scores_.append((start_index, score))
        return score


class DTW:
    """ The function class for dynamic time warping measure

    ----------
    method : string, optional (default='L2' )
        The distance measure to derive DTW.
        Avaliable "L2", "L1", and custom
    Attributes
    ----------
    decision_scores_ : numpy array of shape (n_samples,)
        The outlier scores of the training data.
        The higher, the more abnormal. Outliers tend to have higher
        scores. This value is available once the detector is
        fitted.
    detector: Object classifier
        the anomaly detector that is used
    """
    def __init__(self, method = 'L2'):
        self.decision_scores_  = []
        if type(method) == str:
            if method == 'L1':
                distance = lambda x, y: abs(x-y)
            elif method == 'L2':
                distance = lambda x, y: (x-y)**2
        else:
            distance = method
        self.distance = distance
    def set_param(self):
        '''update the parameters with the detector that is used 
        since the FFT measure doens't need the attributes of detector
        or characteristics of X_train, the process is omitted. 
        '''

        return self

    def measure(self, X1, X2, start_index):
        """Obtain the SSA similarity score.
        Parameters
        ----------
        X1 : numpy array of shape (n, )
            the reference timeseries
        X2 : numpy array of shape (n, )
            the tested timeseries
        index: int, 
        current index for the subseqeuence that is being measured 
        Returns
        -------
        score: float, the higher the more dissimilar are the two curves 
        """       
        distance = self.distance
        X1 = np.array(X1)
        X2 = np.array(X2)
        
        value = 1
        if len(X1)==0:
            value =0
            X1= np.zeros(5)
            X2 = X1
        M = np.zeros((len(X1), len(X2)))
        for index_i in range(len(X1)):
            for index_j in range(len(X1) - index_i):
                L = []
                i = index_i
                j = index_i + index_j
                D = distance(X1[i], X2[j])
                try:
                    L.append(M[i-1, j-1])
                except:
                    L.append(np.inf)
                try:
                    L.append(M[i, j-1])
                except:
                    L.append(np.inf)
                try:
                    L.append(M[i-1, j])
                except:
                    L.append(np.inf)
                D += min(L)
                M[i,j] = D
                if i !=j:
                    L = []
                    j = index_i
                    i = index_i + index_j
                    D = distance(X1[i], X2[j])
                    try:
                        L.append(M[i-1, j-1])
                    except:
                        L.append(np.inf)
                    try:
                        L.append(M[i, j-1])
                    except:
                        L.append(np.inf)
                    try:
                        L.append(M[i-1, j])
                    except:
                        L.append(np.inf)
                    D += min(L)
                    M[i,j] = D
        
        score = M[len(X1)-1, len(X1)-1]/len(X1)
        if value == 0:
            score = 0
        self.decision_scores_.append((start_index, score))
        return score


class EDRS:
    """ The function class for edit distance on real sequences 

    ----------
    method : string, optional (default='L2' )
        The distance measure to derive DTW.
        Avaliable "L2", "L1", and custom
    ep: float, optiona (default = 0.1)
        the threshold value to decide Di_j
    vot : boolean, optional (default = False)
        whether to adapt a chaging votilities estimaed by garch
        for ep at different windows. 
    Attributes
    ----------
    decision_scores_ : numpy array of shape (n_samples,)
        The outlier scores of the training data.
        The higher, the more abnormal. Outliers tend to have higher
        scores. This value is available once the detector is
        fitted.
    detector: Object classifier
        the anomaly detector that is used
    """
    def __init__(self, method = 'L1', ep = False, vol = False):
        self.decision_scores_  = []
        if type(method) == str:
            if method == 'L1':
                distance = lambda x, y: abs(x-y)
        else:
            distance = method
        self.distance = distance
        self.ep = ep
        self.vot = vol
    def set_param(self):
        '''update the ep based on the votalitiy of the model 
        '''
        estimation = np.array(self.detector.estimation )
        initial = self.detector.n_initial_
        X = np.array(self.detector.X_train_)
        self.initial = initial
        residual = estimation[initial:] - X[initial:]
        # number = len(residual)
        # var = (np.sum(np.square(residual))/(number - 1))**0.5
        vot = self.vot
        if vot == False:
            var = np.var(residual)
        else:
            model = arch_model(10 * residual, mean='Constant', vol='garch', p=1, q=1)
            model_fit = model.fit(disp='off')
            var = model_fit.conditional_volatility/10
            
        if self.ep == False:
            self.ep =  3 * (np.sum(np.square(residual))/(len(residual) - 1))**0.5
        else: 
            self.ep = self.ep
        
        return self

    def measure(self, X1, X2, start_index):
        """Obtain the SSA similarity score.
        Parameters
        ----------
        X1 : numpy array of shape (n, )
            the reference timeseries
        X2 : numpy array of shape (n, )
            the tested timeseries
        index: int, 
        current index for the subseqeuence that is being measured 
        Returns
        -------
        score: float, the higher the more dissimilar are the two curves 
        """       
        distance = self.distance
        X1 = np.array(X1)
        X2 = np.array(X2)
        vot = self.vot

        if vot == False:
            ep = self.ep
        else:
            try:
                ep = self.ep[start_index - self.initial]
            except:
                #sometime start_index is the length of the number 
                ep = 0
        value = 1
        if len(X1)==0:
            value =0
            X1= np.zeros(5)
            X2 = X1
        M = np.zeros((len(X1), len(X2)))
        M[:, 0] = np.arange(len(X1))
        M[0, :] = np.arange(len(X1))
        for index_i in range(1, len(X1)):
            for index_j in range(len(X1) - index_i):

                L = []
                i = index_i
                j = index_i + index_j
                D = distance(X1[i], X2[j])
                if D < ep:
                    M[i, j]= M[i-1, j-1]
                else:
                    try:
                        L.append(M[i-1, j-1])
                    except:
                        L.append(np.inf)
                    try:
                        L.append(M[i, j-1])
                    except:
                        L.append(np.inf)
                    try:
                        L.append(M[i-1, j])
                    except:
                        L.append(np.inf)
                    M[i,j] = 1 + min(L)
                if i !=j:
                    L = []
                    j = index_i
                    i = index_i + index_j
                    D = distance(X1[i], X2[j])
                    if D < ep:
                        M[i, j]= M[i-1, j-1]
                    else: 
                        try:
                            L.append(M[i-1, j-1])
                        except:
                            L.append(np.inf)
                        try:
                            L.append(M[i, j-1])
                        except:
                            L.append(np.inf)
                        try:
                            L.append(M[i-1, j])
                        except:
                            L.append(np.inf)
                        M[i,j] = 1 + min(L)

        score = M[len(X1)-1, len(X1)-1]/len(X1)
        if value == 0:
            score = 0
        self.decision_scores_.append((start_index, score))
        return score

class TWED:
    """ Function class for Time-warped edit distance(TWED) measure

    ----------
    method : string, optional (default='L2' )
        The distance measure to derive DTW.
        Avaliable "L2", "L1", and custom
    gamma: float, optiona (default = 0.1)
        mismatch penalty
    v : float, optional (default = False)
        stifness parameter
    Attributes
    ----------
    decision_scores_ : numpy array of shape (n_samples,)
        The outlier scores of the training data.
        The higher, the more abnormal. Outliers tend to have higher
        scores. This value is available once the detector is
        fitted.
    detector: Object classifier
        the anomaly detector that is used
    """
    def __init__(self, gamma = 0.1, v = 0.1):
        self.decision_scores_  = []

        self.gamma = gamma
        self.v = v
    def set_param(self):
        '''No need'''     
        return self
    
    def measure(self, A, B, start_index):
        """Obtain the SSA similarity score.
        Parameters
        ----------
        X1 : numpy array of shape (n, )
            the reference timeseries
        X2 : numpy array of shape (n, )
            the tested timeseries
        index: int, 
        current index for the subseqeuence that is being measured 
        Returns
        -------
        score: float, the higher the more dissimilar are the two curves 
        """    
        #code modifed from wikipedia
        Dlp = lambda x,y: abs(x-y)
        timeSB = np.arange(1,len(B)+1)
        timeSA = np.arange(1,len(A)+1)
        nu = self.v
        _lambda = self.gamma
        # Reference :
        #    Marteau, P.; F. (2009). "Time Warp Edit Distance with Stiffness Adjustment for Time Series Matching".
        #    IEEE Transactions on Pattern Analysis and Machine Intelligence. 31 (2): 306–318. arXiv:cs/0703033
        #    http://people.irisa.fr/Pierre-Francois.Marteau/

        # Check if input arguments
        if len(A) != len(timeSA):
            print("The length of A is not equal length of timeSA")
            return None, None
    
        if len(B) != len(timeSB):
            print("The length of B is not equal length of timeSB")
            return None, None

        if nu < 0:
            print("nu is negative")
            return None, None

        # Add padding
        A = np.array([0] + list(A))
        timeSA = np.array([0] + list(timeSA))
        B = np.array([0] + list(B))
        timeSB = np.array([0] + list(timeSB))

        n = len(A)
        m = len(B)
        # Dynamical programming
        DP = np.zeros((n, m))

        # Initialize DP Matrix and set first row and column to infinity
        DP[0, :] = np.inf
        DP[:, 0] = np.inf
        DP[0, 0] = 0

        # Compute minimal cost
        for i in range(1, n):
            for j in range(1, m):
                # Calculate and save cost of various operations
                C = np.ones((3, 1)) * np.inf
                # Deletion in A
                C[0] = (
                    DP[i - 1, j]
                    + Dlp(A[i - 1], A[i])
                    + nu * (timeSA[i] - timeSA[i - 1])
                    + _lambda
                )
                # Deletion in B
                C[1] = (
                    DP[i, j - 1]
                    + Dlp(B[j - 1], B[j])
                    + nu * (timeSB[j] - timeSB[j - 1])
                    + _lambda
                )
                # Keep data points in both time series
                C[2] = (
                    DP[i - 1, j - 1]
                    + Dlp(A[i], B[j])
                    + Dlp(A[i - 1], B[j - 1])
                    + nu * (abs(timeSA[i] - timeSB[j]) + abs(timeSA[i - 1] - timeSB[j - 1]))
                )
                # Choose the operation with the minimal cost and update DP Matrix
                DP[i, j] = np.min(C)
        distance = DP[n - 1, m - 1]
        self.M = DP
        self.decision_scores_.append((start_index, distance))
        return distance