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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 Load necessary packages
Step1:  Define functions for filtering, moving averages, and normalizing data
Step2:  Read bandwidth and rain/temperature data and normalize them
Step3:  Smoothing data (11-year running average)
Step4:  Calculate correlation and p-values with considering autocorrelation, and the autocorrelations (coef)
Step5:  Check the correlation results
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Python Code:
%matplotlib inline
from scipy import interpolate
from scipy import special
from scipy.signal import butter, lfilter, filtfilt
import matplotlib.pyplot as plt
import numpy as np
from numpy import genfromtxt
from nitime import algorithms as alg
from nitime import utils
from scipy.stats import t
import pandas as pd
Explanation: Load necessary packages
End of explanation
def butter_lowpass(cutoff, fs, order=3):
    nyq = 0.5 * fs
    normal_cutoff = cutoff / nyq 
    b, a = butter(order, normal_cutoff, btype='low', analog=False)
    return b, a
def filter(x, cutoff, axis, fs=1.0, order=3):
    b, a = butter_lowpass(cutoff, fs, order=order)
    y = filtfilt(b, a, x, axis=axis)
    return y
def movingaverage(interval, window_size):
    window = np.ones(int(window_size))/float(window_size)
    return np.convolve(interval, window, 'valid')
def owncorr(x,y,n):
    x_ano=np.ma.anomalies(x)
    x_sd=np.sum(x_ano**2,axis=0)
    y_ano=np.ma.anomalies(y)
    y_sd=np.sum(y_ano**2,axis=0)
    nomi = np.dot(x_ano,y_ano)
    corr = nomi/np.sqrt(np.dot(x_sd[None],y_sd[None]))
    # When using AR_est_YW, we should substract mean from
    # time series first
    x_coef, x_sigma = alg.AR_est_YW (x_ano, 1)
    y_coef, y_sigma = alg.AR_est_YW (y_ano, 1)
    
    if x_coef > 1:
        eps = np.spacing(1.0)
        x_coef = 1.0 - eps**(1/4)
    elif x_coef < 0:
        x_coef = 0.0
        
    if y_coef > 1:
        eps = np.spacing(1.0)
        y_coef = 1.0 - eps**(1/4)
    elif y_coef < 0:
        y_coef = 0.0
        
    neff = n*(1-x_coef*y_coef)/(1+x_coef*y_coef)
    if neff <3:
        neff = 3
    
    coef = []
    coef.append(x_coef)
    coef.append(y_coef)
    tval = corr/np.sqrt(1-corr**2)*np.sqrt(neff-2)
    pval = t.sf(abs(tval),neff-2)*2
    
    return corr,pval,coef
def gaussianize(X):
    n = X.shape[0]
    #p = X.shape[1]
    Xn = np.empty((n,))
    Xn[:] = np.NAN
    nz = np.logical_not(np.isnan(X))
    index = np.argsort(X[nz])
    rank = np.argsort(index)
    CDF = 1.*(rank+1)/(1.*n) -1./(2*n)
    Xn[nz] = np.sqrt(2)*special.erfinv(2*CDF -1)
    return Xn
Explanation: Define functions for filtering, moving averages, and normalizing data
End of explanation
data = genfromtxt('data/scotland.csv', delimiter=',')
bandw = data[0:115,4] # band width (1879-1993), will be correlated with T/P
bandwl = data[3:129,4] # band width (1865-1990), will be correlation with winter NAO
bandwn = gaussianize(bandw) #normalized band width
bandwln = gaussianize(bandwl) #normalized band width
rain = genfromtxt('data/Assynt_P.txt') #precipitaiton
temp = genfromtxt('data/Assynt_T.txt') #temperature
wnao = genfromtxt('data/wnao.txt') #winter NAO
wnao = wnao[::-1]
rainn = gaussianize(rain)
tempn = gaussianize(temp)
#calculate the ratio of temperature over precipitation
ratio = temp/rain
ration = gaussianize(ratio)
Explanation: Read bandwidth and rain/temperature data and normalize them
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bandw_fil = movingaverage(bandw, 11)
bandwn_fil = movingaverage(bandwn, 11)
bandwl_fil = movingaverage(bandwl, 11)
rain_fil = movingaverage(rain, 11)
rainn_fil = movingaverage(rainn, 11)
ratio_fil = movingaverage(ratio, 11)
wnao_fil = movingaverage(wnao, 11)
Explanation: Smoothing data (11-year running average)
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corr_ratio,pval_ratio,coef = owncorr(bandw_fil,ratio_fil,115) #correlation between smoothed bandwidth and ratio
corr_nao,pval_nao,coef_nao = owncorr(bandwl_fil,wnao_fil,126) #correlation between smoothed bandwidth and winter NAO
corr_n,pval_n,coef_n = owncorr(bandwn,ration,115) #correlation between normalized bandwidth and ratio
corr_naon,pval_naon,coef_naon = owncorr(bandwln,wnao,126) #correlation between normalized bandwidtha and winter NAO
Explanation: Calculate correlation and p-values with considering autocorrelation, and the autocorrelations (coef)
End of explanation
print(corr_ratio)
print(pval_ratio)
print(coef)
print(corr_nao)
print(pval_nao)
print(coef_nao)
print(corr_n)
print(pval_n)
print(coef_n)
print(corr_naon)
print(pval_naon)
print(coef_naon)
Explanation: Check the correlation results
End of explanation
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