In [1]:

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

In [2]:

df=pd.read_excel('https://o2.eas.gatech.edu/data/ATL_MonMeanTemp_1879_2020.xls',skiprows=1)
df # display the dataFrame

Out[2]:

	Year	JAN	FEB	MAR	APR	MAY	JUN	JUL	AUG	SEP	OCT	NOV	DEC
0	1879	44.3	43.7	57.6	58.9	69.8	74.9	79.6	73.9	68.7	64.3	53.9	51.5
1	1880	54.3	51.4	55.5	63.4	71.0	76.5	79.2	76.8	69.4	60.8	47.2	42.1
2	1881	40.1	46.6	49.1	59.1	70.8	77.8	81.1	78.8	75.5	67.2	52.6	49.6
3	1882	48.8	52.4	57.4	64.8	66.1	76.6	75.9	75.8	71.7	65.8	51.1	41.4
4	1883	43.0	50.2	50.2	61.6	67.4	76.4	80.1	76.2	71.3	66.0	53.7	48.0
...	...	...	...	...	...	...	...	...	...	...	...	...	...
137	2016	42.4	48.5	60.5	64.1	71.1	81.0	83.4	82.7	78.8	69.6	58.9	48.9
138	2017	52.0	56.1	57.7	67.8	71.1	76.6	81.2	79.2	73.8	65.7	56.2	46.8
139	2018	40.4	57.3	53.5	59.8	74.8	79.8	80.8	79.7	81.0	67.4	50.3	48.6
140	2019	45.9	53.8	55.4	65.3	76.4	77.7	82.2	82.5	82.4	68.4	52.5	51.2
141	2020	49.2	49.9	61.6	61.6	68.9	76.8	82.1	80.9	73.8	67.3	58.9	46.1

142 rows × 13 columns

In [3]:

MonMeanTemp=df.to_numpy()
X=MonMeanTemp[:,0]    # X is Year
Y=MonMeanTemp[:,8]    # Y is August temperature

In [4]:

# linear regression using pseudo inverse OR left divide
N=np.size(X)
m=21
A=np.zeros((N,m))
z=(X-np.mean(X))/np.std(X) # z-transform x axis 
for n in np.arange(0,m):
    A[:,n]=z**n

#x = np.linalg.lstsq(A,Aug,rcond=None)[0]
Apinv=np.linalg.pinv(A)
#Apinv=np.linalg.inv(A.T@A)@A.T
x=Apinv@Y

#polyfit 
C=np.polyfit(z,Y,m-1)

In [5]:

# calculate Yest
Yest=A@x
Yest2=np.poly1d(C)

# plot the line
plt.plot(X,Y)
plt.plot(X,Yest)
plt.plot(X,Yest2(z),'--')
plt.xlabel('time')
plt.ylabel('Atlanta August temperature, deg F')
plt.legend(['Original data','pinv','polyfit'])
plt.show()

In [6]:

# 10th order polynomial fit
C=np.polyfit(z,Y,10)
yest=np.poly1d(C)
plt.plot(X,Y,'-')
plt.plot(X,yest(z),'--')
plt.xlabel('time')
plt.ylabel('Atlanta August temperature, deg F')
plt.show()

In [7]:

from scipy import signal

In [8]:

m=5
Ym = signal.convolve(Y,np.ones(m),'valid')/m

In [9]:

hm=2
plt.plot(X,Y)
plt.plot(X[hm:-hm],Ym)
plt.xlabel('time')
plt.ylabel('Atlanta August temperature, deg F')
plt.show()

In [10]:

Yhat = np.fft.fft(Y)   # get Fourier coefficients
N    = np.size(Y)      
freq = np.fft.fftfreq(N) # get frequency axis

In [11]:

pd=1/freq      # calculate the period
mask= (pd < 5) # logical index with period < 5 yr
Yhat[mask]=0
Ylf=np.fft.ifft(Yhat)

<ipython-input-11-15a6544ffb10>:1: RuntimeWarning: divide by zero encountered in true_divide
  pd=1/freq      # calculate the period

In [12]:

plt.plot(X,Y)
plt.plot(X[hm:-hm],Ym)
plt.plot(X,np.real(Ylf))
plt.xlabel('time')
plt.ylabel('Atlanta August temperature, deg F')
plt.legend(['Original data','5-yr running mean','5-yr FFT filter'])
plt.show()

In [13]:

import matplotlib.image as mpimg

In [14]:

img = mpimg.imread('flower.png')  # get the picture of a flower
img0=img[:,:,0] # just extract one color
plt.imshow(img0,cmap='gray')
plt.show()
N=np.shape(img0)
print('the size of image array is ',str(N))

the size of image array is  (1092, 1122)

In [15]:

# apply fft and retain low frequency component only
imghat=np.fft.fft2(img0)
M=50 # M lowest frequencies only
iM=imghat
iM[M:-M,:]=0 # truncate rows
iM[:,M:-M]=0 # truncate columns
img1=np.fft.ifft2(iM)
plt.imshow(np.real(img1),cmap='gray')
print('the image compression is '+str(100-4*M**2/(N[0]*N[1])*100)+' percent')
plt.show()

the image compression is 99.18382271323448 percent