################
# m1_Bisquare = RLM(data.endog, data.exog, M=norms.TukeyBiweight())
# results_Bisquare1 = m1_Bisquare.fit()
# m2_Bisquare = RLM(data.endog, data.exog, M=norms.TukeyBiweight())
# results_Bisquare2 = m2_Bisquare.fit(cov="H2")
# m3_Bisquare = RLM(data.endog, data.exog, M=norms.TukeyBiweight())
# results_Bisquare3 = m3_Bisquare.fit(cov="H3")
##############################################
# Huber's Proposal 2 scaling #
##############################################
################
### Huber'sT ###
################
m1_Huber_H = RLM(data.endog, data.exog, M=norms.HuberT())
results_Huber1_H = m1_Huber_H.fit(scale_est=scale.HuberScale())
# m2_Huber_H
# m3_Huber_H
# m4 = RLM(data.endog, data.exog, M=norms.HuberT())
# results4 = m1.fit(scale_est="Huber")
# m5 = RLM(data.endog, data.exog, M=norms.Hampel())
# results5 = m2.fit(scale_est="Huber")
# m6 = RLM(data.endog, data.exog, M=norms.TukeyBiweight())
# results6 = m3.fit(scale_est="Huber")
# print """Least squares fit
#%s
#Huber Params, t = 2.
#%s
#Ramsay's E Params
def smafit(X0,
Y0,
W0=None,
cl=0.95,
intercept=True,
robust=False,
rmethod='FastMCD'):
"""Standard Major-Axis (SMA) line fitting
Calculate standard major axis, aka reduced major axis, fit to
data X and Y. The main advantage of this over ordinary least squares is
that the best fit of Y to X will be the same as the best fit of X to Y.
The fit equations and confidence intervals are implemented following
Warton et al. (2006). Robust fits use the FastMCD covariance estimate
from Rousseeuw and Van Driessen (1999). While there are many alternative
robust covariance estimators (e.g. other papers by D.I. Warton using M-estimators),
the FastMCD algorithm is default in Matlab. When the standard error or
uncertainty of each point is known, then weighted SMA may be preferrable to
robust SMA. The conventional choice of weights for each point i is
W_i = 1 / ( var(X_i) + var(Y_i) ), where var() is the variance
(squared standard error).
References
Warton, D. I., Wright, I. J., Falster, D. S. and Westoby, M.:
Bivariate line-fitting methods for allometry, Biol. Rev., 81(02), 259,
doi:10.1017/S1464793106007007, 2006.
Rousseeuw, P. J. and Van Driessen, K.: A Fast Algorithm for the Minimum
Covariance Determinant Estimator, Technometrics, 41(3), 1999.
Parameters
----------
X, Y : array_like
Input values, Must have same length.
W : optional array of weights for each X-Y point, typically W_i = 1/(var(X_i)+var(Y_i))
cl : float (default = 0.95)
Desired confidence level for output.
intercept : boolean (default=True)
Specify if the fitted model should include a non-zero intercept.
The model will be forced through the origin (0,0) if intercept=False.
robust : boolean (default=False)
Use statistical methods that are robust to the presence of outliers
rmethod: string (default='FastMCD')
Method for calculating robust variance and covariance. Options:
'MCD' or 'FastMCD' for Fast MCD
'Huber' for Huber's T: reduce, not eliminate, influence of outliers
'Biweight' for Tukey's Biweight: reduces then eliminates influence of outliers
Returns
-------
Slope : float
Slope or Gradient of Y vs. X
Intercept : float
Y intercept.
ste_grad : float
Standard error of gradient estimate
ste_int : float
standard error of intercept estimate
ci_grad : [float, float]
confidence interval for gradient at confidence level cl
ci_int : [float, float]
confidence interval for intercept at confidence level cl
"""
import numpy as np
import scipy.stats as stats
from sklearn.covariance import MinCovDet
import statsmodels.formula.api as smf
import statsmodels.robust.norms as norms
# Make sure arrays have the same length
assert (len(X0) == len(Y0)), 'Arrays X and Y must have the same length'
if (W0 != None):
assert (
len(W0) == len(X0)), 'Array W must have the same length as X and Y'
# Make sure cl is within the range 0-1
assert (cl < 1), 'cl must be less than 1'
assert (cl > 0), 'cl must be greater than 0'
if (W0 == None):
W0 = np.zeros_like(X0) + 1
# Drop any NaN elements of X or Y
# Infinite values are allowed but will make the result undefined
idx = ~np.logical_or(np.isnan(X0), np.isnan(Y0))
X = X0[idx]
Y = Y0[idx]
W = W0[idx]
# Number of observations
N = len(X)
# Degrees of freedom for the model
if (intercept):
dfmod = 2
else:
dfmod = 1
# Choose whether to use methods robust to outliers
if (robust):
# Choose the robust method
if ((rmethod.lower() == 'mcd') or (rmethod.lower() == 'fastmcd')):
# FAST MCD
if (not intercept):
# intercept=False could possibly be supported by calculating
# using mcd.support_ as weights in an explicit variance/covariance calculation
raise NotImplementedError(
'FastMCD method only supports SMA with intercept')
# Fit robust model of mean and covariance
mcd = MinCovDet().fit(np.array([X, Y]).T)
# Robust mean
Xmean = mcd.location_[0]
Ymean = mcd.location_[1]
# Robust variance of X, Y
Vx = mcd.covariance_[0, 0]
Vy = mcd.covariance_[1, 1]
# Robust covariance
Vxy = mcd.covariance_[0, 1]
# Number of observations used in mean and covariance estimate
# excludes observations marked as outliers
N = mcd.support_.sum()
elif ((rmethod.lower() == 'biweight') or (rmethod.lower() == 'huber')):
# Tukey's Biweight and Huber's T
if (rmethod.lower() == 'biweight'):
norm = norms.TukeyBiweight()
else:
norm = norms.HuberT()
# Get weights for downweighting outliers
# Fitting a linear model the easiest way to get these
# Options include "TukeyBiweight" (totally removes large deviates)
# "HuberT" (linear, not squared weighting of large deviates)
rweights = smf.rlm('y~x+1', {'x': X, 'y': Y}, M=norm).fit().weights
# Sum of weight and weights squared, for convienience
rsum = np.sum(rweights)
rsum2 = np.sum(rweights**2)
# Mean
Xmean = np.sum(X * rweights) / rsum
Ymean = np.sum(Y * rweights) / rsum
# Force intercept through zero, if requested
if (not intercept):
Xmean = 0
Ymean = 0
# Variance & Covariance
Vx = np.sum((X - Xmean)**2 * rweights**2) / rsum2
Vy = np.sum((Y - Ymean)**2 * rweights**2) / rsum2
Vxy = np.sum((X - Xmean) * (Y - Ymean) * rweights**2) / rsum2
# Effective number of observations
N = rsum
else:
raise NotImplementedError(
"smafit.py hasn't implemented rmethod={:%s}".format(rmethod))
else:
if (intercept):
wsum = np.sum(W)
# Average values
Xmean = np.sum(X * W) / wsum
Ymean = np.sum(Y * W) / wsum
# Covariance matrix
cov = np.cov(X, Y, ddof=1, aweights=W**2)
# Variance
Vx = cov[0, 0]
Vy = cov[1, 1]
# Covariance
Vxy = cov[0, 1]
else:
# Force the line to pass through origin by setting means to zero
Xmean = 0
Ymean = 0
wsum = np.sum(W)
# Sum of squares in place of variance and covariance
Vx = np.sum(X**2 * W) / wsum
Vy = np.sum(Y**2 * W) / wsum
Vxy = np.sum(X * Y * W) / wsum
# Standard deviation
Sx = np.sqrt(Vx)
Sy = np.sqrt(Vy)
# Correlation coefficient (equivalent to np.corrcoef()[1,0] for non-robust cases)
R = Vxy / np.sqrt(Vx * Vy)
#############
# SLOPE
Slope = np.sign(R) * Sy / Sx
# Standard error of slope estimate
ste_slope = np.sqrt(1 / (N - dfmod) * Sy**2 / Sx**2 * (1 - R**2))
# Confidence interval for Slope
B = (1 - R**2) / (N - dfmod) * stats.f.isf(1 - cl, 1, N - dfmod)
ci_grad = Slope * (np.sqrt(B + 1) + np.sqrt(B) * np.array([-1, +1]))
#############
# INTERCEPT
if (intercept):
Intercept = Ymean - Slope * Xmean
# Standard deviation of residuals
# New Method: Formula from smatr R package (Warton)
# This formula avoids large residuals of outliers when using robust=True
Sr = np.sqrt(
(Vy - 2 * Slope * Vxy + Slope**2 * Vx) * (N - 1) / (N - dfmod))
# OLD METHOD
# Standard deviation of residuals
#resid = Y - (Intercept + Slope * X )
# Population standard deviation of the residuals
#Sr = np.std( resid, ddof=0 )
# Standard error of the intercept estimate
ste_int = np.sqrt(Sr**2 / N + Xmean**2 * ste_slope**2)
# Confidence interval for Intercept
tcrit = stats.t.isf((1 - cl) / 2, N - dfmod)
ci_int = Intercept + ste_int * np.array([-tcrit, tcrit])
else:
# Set Intercept quantities to zero
Intercept = 0
ste_int = 0
ci_int = np.array([0, 0])
return Slope, Intercept, ste_slope, ste_int, ci_grad, ci_int