Basic EDM Examples

pyEDM Examples¶

In [1]:

Copied!

from pyEDM import *
import matplotlib.pyplot as plt
plt.rcParams["figure.figsize"] = [6, 3.5]
from pyEDM import *
import matplotlib.pyplot as plt
plt.rcParams["figure.figsize"] = [6, 3.5]

In [2]:

Copied!

tentmap      = sampleData['TentMap']
tentmapNoise = sampleData['TentMapNoise']
tentmap      = sampleData['TentMap']
tentmapNoise = sampleData['TentMapNoise']

In [3]:

Copied!

plt.plot( tentmap['TentMap'][1:40] );
plt.plot( tentmap['TentMap'][1:40] );

No description has been provided for this image

Evaluate optimal embedding dimension (via Simplex)¶

In [4]:

Copied!

EmbedDimension( dataFrame = tentmap, lib = "1 100", pred = "201 500", 
                columns = "TentMap", target = "TentMap" );
EmbedDimension( dataFrame = tentmap, lib = "1 100", pred = "201 500", 
                columns = "TentMap", target = "TentMap" );

Optimal embedding dimension is E = 2

Evaluate decay in prediction versus `Tp`¶

In [5]:

Copied!

PredictInterval( dataFrame = tentmap, lib = "1 100", pred = "251 500",
                 columns = "TentMap", target = "TentMap", E = 2 );
PredictInterval( dataFrame = tentmap, lib = "1 100", pred = "251 500",
                 columns = "TentMap", target = "TentMap", E = 2 );

Decay in predictability as forecast interval increases is a possible indication of chaotic dynamics and nonlinearity provided there is insignificant serial correlation.

Simplex prediction¶

In [6]:

Copied!

Simplex( dataFrame = tentmapNoise, lib = [1,500], pred = [501, 550], 
         columns = "TentMap", target = "TentMap", E = 2, showPlot = True );
Simplex( dataFrame = tentmapNoise, lib = [1,500], pred = [501, 550], 
         columns = "TentMap", target = "TentMap", E = 2, showPlot = True );

Simplex Prediction using multivariate embedding¶

The block_3sp data frame is used directly as an embedding: embedded = True

In [7]:

Copied!

block_3sp = sampleData['block_3sp']
block_3sp.head(3)
block_3sp = sampleData['block_3sp']
block_3sp.head(3)

Out[7]:

	time	x_t	x_t-1	x_t-2	y_t	y_t-1	y_t-2	z_t	z_t-1	z_t-2
0	3	-1.917685	1.244882	-0.741863	-0.113188	1.488887	-1.268104	1.535239	-0.481583	-1.863980
1	4	-0.962318	-1.917685	1.244882	-1.106779	-0.113188	1.488887	-1.492956	1.535239	-0.481583
2	5	1.331875	-0.962318	-1.917685	2.385041	-1.106779	-0.113188	-1.119476	-1.492956	1.535239

In [8]:

Copied!

Simplex( dataFrame = block_3sp, lib = [1,99], pred = [100,195], 
         columns = "x_t y_t z_t", target = "x_t",
         embedded = True, showPlot = True );
Simplex( dataFrame = block_3sp, lib = [1,99], pred = [100,195], 
         columns = "x_t y_t z_t", target = "x_t",
         embedded = True, showPlot = True );

Evaluate optimal SMap theta parameter indicating state-dependence and extent of nonlinearity¶

In [9]:

Copied!

PredictNonlinear( dataFrame = tentmapNoise, lib = [1,100], pred = [201,500],
                  columns = "TentMap", target = "TentMap", E = 2 );
PredictNonlinear( dataFrame = tentmapNoise, lib = [1,100], pred = [201,500],
                  columns = "TentMap", target = "TentMap", E = 2 );

SMap theta parameter in the range of [2-3] provides optimal resolution. The non-zero theta peak of predictability indicates nonlinearity.

SMap Prediction¶

SMap Prediction multivariate embedding¶

Note SMap multivariate prediction requires embedded = True

In [11]:

Copied!

SMap( dataFrame = sampleData[ 'circle' ],
      lib = [1,100], pred = [110,190], theta = 4, E = 2, 
      embedded = True, columns = "x y", target = "x", showPlot = True );
SMap( dataFrame = sampleData[ 'circle' ],
      lib = [1,100], pred = [110,190], theta = 4, E = 2, 
      embedded = True, columns = "x y", target = "x", showPlot = True );

Analytical solution of the SMap coefficients are C0 = 0; ∂x/∂x = 0.998; ∂x/∂y = 0.063. Deviations from these values are due to sampling and numerical noise.

Inference of causality via Convergent Cross Mapping (CCM)¶

In [12]:

Copied!

sardine_anchovy_sst = sampleData['sardine_anchovy_sst']
sardine_anchovy_sst[['anchovy','np_sst']].plot(lw=2);
sardine_anchovy_sst = sampleData['sardine_anchovy_sst']
sardine_anchovy_sst[['anchovy','np_sst']].plot(lw=2);

In [13]:

Copied!





CCM( dataFrame = sardine_anchovy_sst, E = 3, 
     columns = "anchovy", target = "np_sst",
     libSizes = [10,70,10], sample = 30, 
     seed = 12345, showPlot = True );
CCM( dataFrame = sardine_anchovy_sst, E = 3, 
     columns = "anchovy", target = "np_sst",
     libSizes = [10,70,10], sample = 30, 
     seed = 12345, showPlot = True );

The CCM result anchovy:np_sst is interpreted as: sea surface temperature (np_sst) influences anchovy population. The reverse influence (np_sst:anchovy) indicates no link consistent with the fact that anchovy population does not influence sea surface temperature.

Multiview Embedding¶

In [14]:

Copied!

MV = Multiview( dataFrame = block_3sp, lib = [1,99], pred = [105,190], E = 3,
                columns = "x_t y_t z_t", target = "x_t", showPlot = True );
MV = Multiview( dataFrame = block_3sp, lib = [1,99], pred = [105,190], E = 3,
                columns = "x_t y_t z_t", target = "x_t", showPlot = True );

In [15]:

Copied!

MV['View']
# Col_i are column indices in block_3sp
MV['View']
# Col_i are column indices in block_3sp

Out[15]:

	Columns	rho	MAE	CAE	RMSE
0	(x_t(t-0), z_t(t-0), x_t(t-1))	0.920810	0.641525	21.372022	0.316416
1	(x_t(t-0), x_t(t-1), y_t(t-2))	0.867700	1.301578	28.330831	0.411275
2	(x_t(t-0), x_t(t-1), x_t(t-2))	0.931867	0.654676	19.581714	0.293402
3	(x_t(t-0), x_t(t-1), z_t(t-1))	0.918613	0.698145	21.217600	0.319911
4	(x_t(t-0), z_t(t-0), z_t(t-2))	0.885830	0.886854	26.062664	0.373766
5	(x_t(t-0), x_t(t-1), y_t(t-1))	0.901727	1.067862	24.110364	0.356719
6	(x_t(t-0), y_t(t-0), z_t(t-2))	0.777399	1.133895	36.041332	0.511584
7	(x_t(t-0), z_t(t-0), x_t(t-2))	0.872370	1.325145	27.139119	0.400680
8	(x_t(t-0), y_t(t-0), z_t(t-0))	0.781226	1.475317	33.309865	0.508688