时间序列之holt-winters（三次指数平滑）

公式：

# import needed packages
#-----------------------import math
import numpy  as np
import pandas as pdfrom sklearn        import linear_model
from scipy.optimize import fmin_l_bfgs_b# bring in the passenger data from HW4 to test the function against R output
#---------------------------------------------------------------------------
tsA=[12.7, 14.2, 9.6, 9.2, 10.3, 9.3, 7.8, 7.7, 7.4, 6.9, 6.7, 6.9, 6.6]
# define main function [holtWinters] to generate retrospective smoothing/predictions
#-----------------------------------------------------------------------------------def holtWinters(ts, p, sp, ahead, mtype, alpha = None, beta = None, gamma = None):'''HoltWinters retrospective smoothing & future period prediction algorithm both the additive and multiplicative methods are implemented and the (alpha, beta, gamma)parameters have to be either all user chosen or all optimized via one-step-ahead prediction MSDinitial (a, b, s) parameter values are calculated with a fixed-period seasonal decomposition and asimple linear regression to estimate the initial level (B0) and initial trend (B1) values@params:- ts[list]:      time series of data to model- p[int]:        period of the time series (for the calculation of seasonal effects)- sp[int]:       number of starting periods to use when calculating initial parameter values- ahead[int]:    number of future periods for which predictions will be generated- mtype[string]: which method to use for smoothing/forecasts ['additive'/'multiplicative']- alpha[float]:  user-specified level  forgetting factor (one-step MSD optimized if None)- beta[float]:   user-specified slope  forgetting factor (one-step MSD optimized if None)- gamma[float]:  user-specified season forgetting factor (one-step MSD optimized if None)@return: - alpha[float]:    chosen/optimal level  forgetting factor used in calculations- beta[float]:     chosen/optimal trend  forgetting factor used in calculations- gamma[float]:    chosen/optimal season forgetting factor used in calculations- MSD[float]:      chosen/optimal Mean Square Deviation with respect to one-step-ahead predictions- params[tuple]:   final (a, b, s) parameter values used for the prediction of future observations- smoothed[list]:  smoothed values (level + trend + seasonal) for the original time series- predicted[list]: predicted values for the next @ahead periods of the time seriessample calls:results = holtWinters(ts, 12, 4, 24, 'additive')results = holtWinters(ts, 12, 4, 24, 'multiplicative', alpha = 0.1, beta = 0.2, gamma = 0.3)'''a, b, s = _initValues(mtype, ts, p, sp)if alpha == None or beta == None or gamma == None:ituning   = [0.1, 0.1, 0.1]ibounds   = [(0,1), (0,1), (0,1)]optimized = fmin_l_bfgs_b(_MSD, ituning, args = (mtype, ts, p, a, b, s[:]), bounds = ibounds, approx_grad = True)alpha, beta, gamma = optimized[0]MSD, params, smoothed = _expSmooth(mtype, ts, p, a, b, s[:], alpha, beta, gamma)predicted = _predictValues(mtype, p, ahead, params)return {'alpha': alpha, 'beta': beta, 'gamma': gamma, 'MSD': MSD, 'params': params, 'smoothed': smoothed, 'predicted': predicted}def _initValues(mtype, ts, p, sp):'''subroutine to calculate initial parameter values (a, b, s) based on a fixed number of starting periods'''initSeries = pd.Series(ts[:p*sp])if mtype == 'additive':# print(p)rawSeason  = initSeries - initSeries.rolling(window = p, min_periods = p, center = True).mean()initSeason = [np.nanmean(rawSeason[i::p]) for i in range(p)]initSeason = pd.Series(initSeason) - np.mean(initSeason)deSeasoned = [initSeries[v] - initSeason[v % p] for v in range(len(initSeries))]else:rawSeason  = initSeries / initSeries.rolling(window = p, min_periods = p, center = True).mean()initSeason = [np.nanmean(rawSeason[i::p]) for i in range(p)]initSeason = pd.Series(initSeason) / math.pow(np.prod(np.array(initSeason)), 1/p)deSeasoned = [initSeries[v] / initSeason[v % p] for v in range(len(initSeries))]lm = linear_model.LinearRegression()lm.fit(pd.DataFrame({'time': [t+1 for t in range(len(initSeries))]}), pd.Series(deSeasoned))return float(lm.intercept_), float(lm.coef_), list(initSeason)def _MSD(tuning, *args):'''subroutine to pass to BFGS optimization to determine the optimal (alpha, beta, gamma) values'''predicted = []mtype     = args[0]ts, p     = args[1:3]Lt1, Tt1  = args[3:5]St1       = args[5][:]alpha, beta, gamma = tuning[:]for t in range(len(ts)):if mtype == 'additive':Lt = alpha * (ts[t] - St1[t % p]) + (1 - alpha) * (Lt1 + Tt1)Tt = beta  * (Lt - Lt1)           + (1 - beta)  * (Tt1)St = gamma * (ts[t] - Lt)         + (1 - gamma) * (St1[t % p])predicted.append(Lt1 + Tt1 + St1[t % p])else:Lt = alpha * (ts[t] / St1[t % p]) + (1 - alpha) * (Lt1 + Tt1)Tt = beta  * (Lt - Lt1)           + (1 - beta)  * (Tt1)St = gamma * (ts[t] / Lt)         + (1 - gamma) * (St1[t % p])predicted.append((Lt1 + Tt1) * St1[t % p])Lt1, Tt1, St1[t % p] = Lt, Tt, Streturn sum([(ts[t] - predicted[t])**2 for t in range(len(predicted))])/len(predicted)def _expSmooth(mtype, ts, p, a, b, s, alpha, beta, gamma):'''subroutine to calculate the retrospective smoothed values and final parameter values for prediction'''smoothed = []Lt1, Tt1, St1 = a, b, s[:]for t in range(len(ts)):if mtype == 'additive':Lt = alpha * (ts[t] - St1[t % p]) + (1 - alpha) * (Lt1 + Tt1)Tt = beta  * (Lt - Lt1)           + (1 - beta)  * (Tt1)St = gamma * (ts[t] - Lt)         + (1 - gamma) * (St1[t % p])smoothed.append(Lt1 + Tt1 + St1[t % p])else:Lt = alpha * (ts[t] / St1[t % p]) + (1 - alpha) * (Lt1 + Tt1)Tt = beta  * (Lt - Lt1)           + (1 - beta)  * (Tt1)St = gamma * (ts[t] / Lt)         + (1 - gamma) * (St1[t % p])smoothed.append((Lt1 + Tt1) * St1[t % p])Lt1, Tt1, St1[t % p] = Lt, Tt, StMSD = sum([(ts[t] - smoothed[t])**2 for t in range(len(smoothed))])/len(smoothed)return MSD, (Lt1, Tt1, St1), smootheddef _predictValues(mtype, p, ahead, params):'''subroutine to generate predicted values @ahead periods into the future'''Lt, Tt, St = paramsif mtype == 'additive':return [Lt + (t+1)*Tt + St[t % p] for t in range(ahead)]else:return [(Lt + (t+1)*Tt) * St[t % p] for t in range(ahead)]# print out the results to check against R output
#------------------------------------------------results = holtWinters(tsA, 4, 4, 5, mtype = 'additive')
# # results = holtWinters(tsA, 12, 4, 24, mtype = 'multiplicative')print("TUNING: ", results['alpha'], results['beta'], results['gamma'], results['MSD'])
# # print('----')
print("FINAL PARAMETERS: ", results['params'])
print("PREDICTED VALUES: ", results['predicted'])

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