About xMCA

xMCA, inspired from EOFs https://github.com/ajdawson/eofs, is Maximum Covariance Analysis (sometimes also called SVD) in xarray.

API Documentation: http://xmca.readthedocs.io/

How to install

via git

git clone https://github.com/Yefee/xMCA.git
cd xMCA
python setup.py install

Future Plan

In next version, A Monte Carlo method will be added for statistical test.

Example 1

MCA analysis for US surface air temperature and SST over the Pacific. This example is taken from https://atmos.washington.edu/~breth/classes/AS552/matlab/lect/html/MCA_PSSTA_USTA.html

from xMCA import xMCA
import xarray as xr
import matplotlib.pyplot as plt
%matplotlib inline

usta = xr.open_dataarray('xMCA/examples/data/USTA.nc').transpose(*['time', 'lat', 'lon'])
usta.name = 'USTA'
print(usta)

<xarray.DataArray 'USTA' (time: 396, lat: 5, lon: 12)>
array([[[-0.450303, -0.734848, ..., -4.270303, -2.69697 ],
        [ 1.066061,  2.691515, ..., -4.947273, -3.330303],
        ...,
        [      nan, -0.342424, ...,       nan,       nan],
        [      nan,       nan, ...,       nan,       nan]],

       [[ 1.524545,  1.370606, ..., -1.430303,  0.048485],
        [ 1.366364,  2.497273, ..., -0.593939, -0.079697],
        ...,
        [      nan,  0.695455, ...,       nan,       nan],
        [      nan,       nan, ...,       nan,       nan]],

       ...,

       [[ 1.077879,  0.630303, ..., -1.262727, -1.496364],
        [ 1.020606,  0.114848, ..., -0.786667, -0.573939],
        ...,
        [      nan,  1.65    , ...,       nan,       nan],
        [      nan,       nan, ...,       nan,       nan]],

       [[ 1.768182,  2.807879, ...,  0.885758,  0.618182],
        [ 1.555152,  3.435152, ..., -0.416667,  0.185152],
        ...,
        [      nan,  0.012121, ...,       nan,       nan],
        [      nan,       nan, ...,       nan,       nan]]])
Coordinates:
  * lat      (lat) float64 47.5 42.5 37.5 32.5 27.5
  * lon      (lon) float64 -122.5 -117.5 -112.5 -107.5 ... -77.5 -72.5 -67.5
  * time     (time) int64 0 1 2 3 4 5 6 7 8 ... 388 389 390 391 392 393 394 395

sstpc = xr.open_dataarray('xMCA/examples/data/SSTPac.nc').transpose(*['time', 'lat', 'lon'])
sstpc.name = 'SSTPC'
print(sstpc)

<xarray.DataArray 'SSTPC' (time: 396, lat: 30, lon: 84)>
[997920 values with dtype=float64]
Coordinates:
  * lat      (lat) int16 -29 -27 -25 -23 -21 -19 -17 ... 17 19 21 23 25 27 29
  * lon      (lon) uint16 124 126 128 130 132 134 ... 280 282 284 286 288 290
  * time     (time) int64 0 1 2 3 4 5 6 7 8 ... 388 389 390 391 392 393 394 395

Decompsition and retrieve the first and second loadings and expansion coefficeints

'''
decomposition, time should be in the first axis
lp is for SSTPC
rp is for USTA
'''

sst_ts = xMCA(sstpc, usta)
sst_ts.solver()
lp, rp = sst_ts.patterns(n=2)
le, re = sst_ts.expansionCoefs(n=2)
frac = sst_ts.covFracs(n=2)
print(frac)

<xarray.DataArray 'frac' (n: 2)>
array([0.407522, 0.391429])
Coordinates:
  * n        (n) int64 0 1
Attributes:
    long_name:  Fractions explained of the covariance matrix between SSTPC an...

fig, (ax1, ax2) = plt.subplots(2, 2, figsize=(12, 5))
lp[0].plot(ax=ax1[0])
le[0].plot(ax=ax1[1])

rp[0].plot(ax=ax2[0])
re[0].plot(ax=ax2[1])

Homogeneous and heterogeneous regression

lh, rh = sst_ts.homogeneousPatterns(n=1)
le, re = sst_ts.heterogeneousPatterns(n=1)

fig, (ax1, ax2) = plt.subplots(2, 2, figsize=(12, 5))
lh[0].plot(ax=ax1[0])
rh[0].plot(ax=ax1[1])

le[0].plot(ax=ax2[0])
re[0].plot(ax=ax2[1])

Two-tailed Student-t test

le, re, lphet, rphet = sst_ts.heterogeneousPatterns(n=1, statistical_test=True)

fig, (ax1, ax2) = plt.subplots(2, 2, figsize=(12, 5))
le[0].plot(ax=ax1[0])
re[0].plot(ax=ax1[1])

# Only plot where p<0.01
lphet[0].where(lphet[0]<0.01).plot(ax=ax2[0])
rphet[0].where(rphet[0]<0.01).plot(ax=ax2[1])

Example 2

EOF analysis for US surface air temperature and SST over the Pacific This example is taken from https://atmos.washington.edu/~breth/classes/AS552/matlab/lect/html/MCA_PSSTA_USTA.html

from xMCA import xMCA
import xarray as xr
import matplotlib.pyplot as plt
%matplotlib inline

sstpc = xr.open_dataarray('data/SSTPac.nc').transpose(*['time', 'lat', 'lon'])
sstpc.name = 'SSTPC'
print(sstpc)

<xarray.DataArray 'SSTPC' (time: 396, lat: 30, lon: 84)>
[997920 values with dtype=float64]
Coordinates:
  * lat      (lat) int16 -29 -27 -25 -23 -21 -19 -17 ... 17 19 21 23 25 27 29
  * lon      (lon) uint16 124 126 128 130 132 134 ... 280 282 284 286 288 290
  * time     (time) int64 0 1 2 3 4 5 6 7 8 ... 388 389 390 391 392 393 394 395

Decompsition and retrieve the first and second loadings and expansion coefficeints

'''
decomposition, time should be in the first axis
lp is for SSTPC
rp is for USTA
'''

sst_ts = xMCA(sstpc, sstpc.rename('SSTPC_copy'))
sst_ts.solver()
lp, _ = sst_ts.patterns(n=2)
le, _ = sst_ts.expansionCoefs(n=2)
frac = sst_ts.covFracs(n=2)
print(frac)

<xarray.DataArray 'frac' (n: 2)>
array([0.873075, 0.04946 ])
Coordinates:
  * n        (n) int64 0 1
Attributes:
    long_name:  Fractions explained of the covariance matrix between SSTPC an...

fig, ax1 = plt.subplots(1, 2, figsize=(12, 5))
lp[0].plot(ax=ax1[0])
le[0].plot(ax=ax1[1])

Regress PC1 to the original SST field

lh, _ = sst_ts.homogeneousPatterns(n=1)

fig, ax1= plt.subplots()
lh[0].plot(ax=ax1)