From 15569a43695f68094c8295b026705ad95e27f59d Mon Sep 17 00:00:00 2001
From: Jeremy Fee <jmfee@usgs.gov>
Date: Tue, 26 Jan 2016 13:31:53 -0700
Subject: [PATCH] Finishing rename from SQDist to SqDist
---
geomagio/algorithm/SqDistAlgorithm.py | 393 ++++++++++++++++++++++++++
1 file changed, 393 insertions(+)
create mode 100644 geomagio/algorithm/SqDistAlgorithm.py
diff --git a/geomagio/algorithm/SqDistAlgorithm.py b/geomagio/algorithm/SqDistAlgorithm.py
new file mode 100644
index 000000000..2905055b5
--- /dev/null
+++ b/geomagio/algorithm/SqDistAlgorithm.py
@@ -0,0 +1,393 @@
+"""Algorithm that produces Solar Quiet (SQ), Secular Variation (SV) and
+ Magnetic Disturbance (DIST).
+
+ Algorithm for producing SQ, SV and DIST.
+ This module implements Holt-Winters exponential smoothing. It predicts
+ an offset-from-zero plus a "seasonal" correction given observations
+ up to time t-1. Each observation's influence on the current prediction
+ decreases exponentially with time according to user-supplied runtime
+ configuration parameters.
+
+ Use of fmin_l_bfgs_b to estimate parameters inspired by Andre Queiroz:
+ https://gist.github.com/andrequeiroz/5888967
+"""
+
+from Algorithm import Algorithm
+from AlgorithmException import AlgorithmException
+import numpy as np
+from scipy.optimize import fmin_l_bfgs_b
+
+
+class SqDistAlgorithm(Algorithm):
+ """Solar Quiet, Secular Variation, and Disturbance algorithm"""
+
+ def __init__(self, alpha=None, beta=None, gamma=None, phi=1,
+ yhat0=None, s0=None, l0=None, b0=None, sigma0=None,
+ zthresh=6, fc=0, hstep=0):
+ self.alpha = alpha
+ self.beta = beta
+ self.gamma = gamma
+ self.phi = phi
+ self.yhat0 = yhat0
+ self.s0 = s0
+ self.l0 = l0
+ self.b0 = b0
+ self.sigma0 = sigma0
+ self.zthresh = zthresh
+ self.fc = fc
+ self.hstep = hstep
+
+ def process(self, yobs, m):
+ pass
+
+ @classmethod
+ def additive(yobs, m, alpha, beta, gamma, phi=1,
+ yhat0=None, s0=None, l0=None, b0=None, sigma0=None,
+ zthresh=6, fc=0, hstep=0):
+ """Primary function for Holt-Winters smoothing/forecasting with
+ damped linear trend and additive seasonal component.
+
+ The adaptive standard deviation (sigma), multiplied by zthresh to
+ determine which observations should be smoothed or ignored, is always
+ updated using the latest error if a valid observation is available.
+ This way, if what seemed a spike in real-time was actually a more
+ permanent baseline shift, the algorithm will adjust to the new baseline
+ once sigma grows enough to accommodate the errors.
+
+ The standard deviation also updates when no obserations are present,
+ but does so according to Hyndman et al (2005) prediction intervals.
+ The result is a sigma that grows over gaps, and for forecasts beyond
+ yobs[-1].
+
+ Parameters
+ ----------
+ yobs : array_like
+ input series to be smoothed/forecast
+ m : int
+ number of "seasons"
+ alpha : float
+ the level smoothing parameter (0<=alpha<=1).
+ beta : float
+ the slope smoothing parameter (0<=beta<=1).
+ gamma : float
+ the seasonal adjustment smoothing parameter (0<=gamma<=1).
+ phi : float
+ the dampening factor for slope (0<=phi<=1)
+ (if None, phi will be estimated; default is 1)
+ yhat0 : array_like
+ initial yhats for hstep>0 (len(yhat0) == hstep)
+ (if None, yhat0 will be set to NaNs)
+ s0 : array_like
+ initial set of seasonal adjustments
+ (if None, default is [yobs[i] - a[0] for i in range(m)])
+ l0 : float
+ initial level (i.e., l(t-hstep))
+ (if None, default is mean(yobs[0:m]))
+ b0 : float
+ initial slope (i.e., b(t-hstep))
+ (if None, default is (mean(yobs[m:2*m]) - mean(yobs[0:m]))/m )
+ sigma0 : float
+ initial standard-deviation estimate (len(sigma0) == hstep+1)
+ (if None, default is [sqrt(var(yobs))] * (hstep+1) )
+ zthresh : int
+ z-score threshold to determine whether yhat is updated by
+ smoothing observations, or by simulation alone; if exceeded,
+ only sigma is updated to reflect latest observation
+ fc : int
+ the number of steps beyond the end of yobs (the available
+ observations) to forecast
+ hstep : int
+ the number of steps ahead to predict yhat[i]
+ which forces an hstep prediction at each time step
+
+ Returns
+ -------
+ yhat : array_like
+ series of smoothed/forecast values (aligned with yobs(t))
+ shat : array_like
+ series of seasonal adjustments (aligned with yobs(t))
+ sigmahat : array_like
+ series of time-varying standard deviations (aligned with yobs(t))
+ yhat0next : array_like
+ use as yhat0 when function called again with new observations
+ s0next : float
+ use as s0 when function called again with new observations
+ l0next : float
+ use as l0 when function called again with new observations
+ b0next : float
+ use as b0 when function called again with new observations
+ sigma0next : float
+ use as sigma0 when function called again with new observations
+ alpha : float
+ optimized alpha (if input alpha is None)
+ beta : float
+ optimized beta (if input beta is None)
+ gamma : float
+ optimized gamma (if input gamma is None)
+ phi : float
+ optimized phi (if input phi is None)
+ """
+
+ if alpha is None:
+ raise AlgorithmException('alpha is required')
+ if beta is None:
+ raise AlgorithmException('beta is required')
+ if gamma is None:
+ raise AlgorithmException('gamma is required')
+ if phi is None:
+ raise AlgorithmException('phi is required')
+
+ # set some default values
+ if l0 is None:
+ l = np.nanmean(yobs[0:int(m)])
+ else:
+ l = l0
+ if not np.isscalar(l0):
+ raise AlgorithmException("l0 must be a scalar")
+
+ if b0 is None:
+ b = (np.nanmean(yobs[m:2 * m]) - np.nanmean(yobs[0:m])) / m
+ b = 0 if np.isnan(b) else b # replace NaN with 0
+ else:
+ b = b0
+ if not np.isscalar(b0):
+ raise AlgorithmException("b0 must be a scalar")
+
+ if yhat0 is None:
+ yhat = [np.nan for i in range(hstep)]
+ else:
+ yhat = list(yhat0)
+ if len(yhat) != hstep:
+ raise AlgorithmException("yhat0 must have length %d" % hstep)
+
+ if s0 is None:
+ s = [yobs[i] - l for i in range(m)]
+ s = [i if ~np.isnan(i) else 0 for i in s] # replace NaNs with 0s
+ else:
+ s = list(s0)
+ if len(s) != m:
+ raise AlgorithmException("s0 must have length %d " % m)
+
+ if sigma0 is None:
+ sigma = [np.sqrt(np.nanvar(yobs))] * (hstep + 1)
+ else:
+ sigma = list(sigma0)
+ if len(sigma) != (hstep + 1):
+ raise AlgorithmException(
+ "sigma0 must have length %d" % (hstep + 1))
+
+ #
+ # Now begin the actual Holt-Winters algorithm
+ #
+
+ # ensure mean of seasonal adjustments is zero by setting first element
+ # of r equal to mean(s)
+ r = [np.nanmean(s)]
+
+ # determine sum(c^2) and phi_(j-1) for hstep "prediction interval"
+ # outside of loop; initialize variables for jstep (beyond hstep)
+ # prediction intervals
+ sumc2_H = 1
+ phiHminus1 = 0
+ for h in range(1, hstep):
+ phiHminus1 = phiHminus1 + phi ** (h - 1)
+ sumc2_H = sumc2_H + (alpha * (1 + phiHminus1 * beta) +
+ gamma * (1 if (h % m == 0) else 0)) ** 2
+ phiJminus1 = phiHminus1
+ sumc2 = sumc2_H
+ jstep = hstep
+
+ # convert to, and pre-allocate numpy arrays
+ yobs = np.array(yobs)
+ sigma = np.concatenate((sigma, np.zeros(yobs.size + fc)))
+ yhat = np.concatenate((yhat, np.zeros(yobs.size + fc)))
+ r = np.concatenate((r, np.zeros(yobs.size + fc)))
+ s = np.concatenate((s, np.zeros(yobs.size + fc)))
+
+ # smooth/simulate/forecast yobs
+ for i in range(len(yobs) + fc):
+ # Update/append sigma for h steps ahead of i following
+ # Hyndman-et-al-2005. This will be over-written if valid
+ # observations exist at step i
+ if jstep == hstep:
+ sigma2 = sigma[i] * sigma[i]
+ sigma[i + hstep + 1] = np.sqrt(sigma2 * sumc2)
+
+ # predict h steps ahead
+ yhat[i + hstep] = l + phiHminus1 * b + s[i + hstep % m]
+
+ # discrepancy between observation and prediction at step i
+ if i < len(yobs):
+ et = yobs[i] - yhat[i]
+ else:
+ et = np.nan
+
+ if (np.isnan(et) or np.abs(et) > zthresh * sigma[i]):
+ # forecast (i.e., update l, b, and s assuming et==0)
+
+ # no change in seasonal adjustments
+ r[i + 1] = 0
+ s[i + m] = s[i]
+
+ # update l before b
+ l = l + phi * b
+ b = phi * b
+
+ if np.isnan(et):
+ # when forecasting, grow sigma=sqrt(var) like a prediction
+ # interval; sumc2 and jstep will be reset with the next
+ # valid observation
+ phiJminus1 = phiJminus1 + phi ** jstep
+ sumc2 = sumc2 + (alpha * (1 + phiJminus1 * beta) +
+ gamma * (1 if (jstep % m == 0) else 0)) ** 2
+ jstep = jstep + 1
+
+ else:
+ # still update sigma using et when et > zthresh * sigma
+ # (and is not NaN)
+ sigma[i + 1] = alpha * np.abs(et) + (1 - alpha) * sigma[i]
+ else:
+ # smooth (i.e., update l, b, and s by filtering et)
+
+ # renormalization could occur inside loop, but we choose to
+ # integrate r, and adjust a and s outside the loop to improve
+ # performance.
+ r[i + 1] = gamma * (1 - alpha) * et / m
+
+ # update and append to s using equation-error formulation
+ s[i + m] = s[i] + gamma * (1 - alpha) * et
+
+ # update l and b using equation-error formulation
+ l = l + phi * b + alpha * et
+ b = phi * b + alpha * beta * et
+
+ # update sigma with et, then reset prediction interval
+ sigma[i + 1] = alpha * np.abs(et) + (1 - alpha) * sigma[i]
+ sumc2 = sumc2_H
+ phiJminus1 = phiHminus1
+ jstep = hstep
+ # endif (np.isnan(et) or np.abs(et) > zthresh * sigma[i])
+
+ # endfor i in range(len(yobs) + fc - hstep)
+
+ r = np.cumsum(r)
+ l = l + r[-1]
+ s = list(np.array(s) - np.hstack((r, np.tile(r[-1], m - 1))))
+
+ return (yhat[:len(yobs) + fc],
+ s[:len(yobs) + fc],
+ sigma[1:len(yobs) + fc + 1],
+ yhat[len(yobs) + fc:],
+ s[len(yobs) + fc:],
+ l,
+ b,
+ sigma[len(yobs) + fc:],
+ alpha,
+ beta,
+ gamma)
+
+ @classmethod
+ def estimate_parameters(yobs, m, alpha=None, beta=None, gamma=None, phi=1,
+ yhat0=None, s0=None, l0=None, b0=None, sigma0=None,
+ zthresh=6, fc=0, hstep=0,
+ alpha0=0.3, beta0=0.1, gamma0=0.1):
+ """Estimate alpha, beta, and gamma parameters based on observed data.
+
+ Parameters
+ ----------
+ yobs : array_like
+ input series to be smoothed/forecast
+ m : int
+ number of "seasons"
+ alpha : float
+ the level smoothing parameter (0<=alpha<=1).
+ beta : float
+ the slope smoothing parameter (0<=beta<=1).
+ gamma : float
+ the seasonal adjustment smoothing parameter (0<=gamma<=1).
+ phi : float
+ the dampening factor for slope (0<=phi<=1)
+ (if None, phi will be estimated; default is 1)
+ yhat0 : array_like
+ initial yhats for hstep>0 (len(yhat0) == hstep)
+ (if None, yhat0 will be set to NaNs)
+ s0 : array_like
+ initial set of seasonal adjustments
+ (if None, default is [yobs[i] - a[0] for i in range(m)])
+ l0 : float
+ initial level (i.e., l(t-hstep))
+ (if None, default is mean(yobs[0:m]))
+ b0 : float
+ initial slope (i.e., b(t-hstep))
+ (if None, default is (mean(yobs[m:2*m]) - mean(yobs[0:m]))/m )
+ sigma0 : float
+ initial standard-deviation estimate (len(sigma0) == hstep+1)
+ (if None, default is [sqrt(var(yobs))] * (hstep+1) )
+ zthresh : int
+ z-score threshold to determine whether yhat is updated by
+ smoothing observations, or by simulation alone; if exceeded,
+ only sigma is updated to reflect latest observation
+ fc : int
+ the number of steps beyond the end of yobs (the available
+ observations) to forecast
+ hstep : int
+ the number of steps ahead to predict yhat[i]
+ which forces an hstep prediction at each time step
+ alpha0 : float
+ initial value for alpha.
+ used only when alpha is None.
+ beta0 : float
+ initial value for beta.
+ used only when beta is None.
+ gamma0 : float
+ initial value for gamma.
+ used only when gamma is None.
+
+ Returns
+ -------
+ alpha : float
+ optimized parameter alpha (if alpha was None).
+ beta : float
+ optimized parameter beta (if beta was None).
+ gamma : float
+ optimized gamma, if gamma was None.
+ rmse : float
+ root-mean-squared-error for data using optimized parameters.
+ """
+ # if alpha/beta/gamma is specified, restrict bounds to "fix" parameter.
+ boundaries = [
+ (alpha, alpha) if alpha is not None else (0, 1),
+ (beta, beta) if beta is not None else (0, 1),
+ (gamma, gamma) if gamma is not None else (0, 1)
+ ]
+ initial_values = np.array([
+ alpha if alpha is not None else alpha0,
+ beta if beta is not None else beta0,
+ gamma if gamma is not None else gamma0
+ ])
+
+ def func(params, *args):
+ """Function that computes root-mean-squared-error based on current
+ alpha, beta, and gamma parameters; as provided by fmin_l_bfgs_b.
+
+ Parameters
+ ----------
+ params: list-like
+ list containing alpha, beta, and gamma parameters to test
+ """
+ # extract parameters to fit
+ alpha, beta, gamma = params
+ # call Holt-Winters with additive seasonality
+ yhat, _, _, _, _, _, _, _, _, _, _ = SqDistAlgorithm.additive(
+ yobs, m,
+ alpha=alpha, beta=beta, gamma=gamma, l0=l0, b0=b0, s0=s0,
+ zthresh=zthresh, hstep=hstep)
+ # compute root-mean-squared-error of predictions
+ error = np.sqrt(np.nanmean(np.square(np.subtract(yobs, yhat))))
+ return error
+
+ parameters = fmin_l_bfgs_b(func, x0=initial_values, args=(),
+ bounds=boundaries, approx_grad=True)
+ alpha, beta, gamma = parameters[0]
+ rmse = parameters[1]
+ return (alpha, beta, gamma, rmse)
--
GitLab