propagation of error

Is there an operation we can use that would overlay and stretch these curves optimally (given they are on different axes), and then provide a metric to quantify the “similarity” between these two curves? Simple x-y correlation loses the time series covariation. Perhaps something like a minimum Euclidian distance? Also, given the raw data are kind of noisy, the raw data are fitted with GAM functions (which are the curves you see below), rather than using the “point to point” trends. However these come with a level of uncertainty (i.e. the 95% confidence intervals, indicated by the area fill). Is there a way of propagating this uncertainty in the “similarity” metric? Perhaps a similarity metric with a +/- value that takes the variation of the GAM functions into account?

Various curves needing to be compared statistically for their degree of similarity.

python code used to make plots. These are geology-geochemistry plots, where depth is treated like a time series, reading from bottom (oldest) to top (youngest). Apologies for how long the code is, I am not an expert coder.

import pandas as pd
import matplotlib.pyplot as plt
import numpy as np
from pygam import LinearGAM, s, f

# Read data
data = pd.read_csv(“Geochemical data.csv”, encoding=”latin1″)
data[“Depth”] = pd.to_numeric(data[“Depth”])

# Columns to convert to numeric
columns_to_convert = [“proxy_1”, “Depth”, “proxy_2″]

# Convert columns to numeric, replacing non-numeric values with NaN
data[columns_to_convert] = data[columns_to_convert].apply(pd.to_numeric, errors=’coerce’)

# Drop rows with NaN values in the specified columns
data = data.dropna(subset=columns_to_convert)

# Filter data by drillcore and remove NaN values
data_filtered = data[(data[‘Drillcore_ID’] == ‘drillcore example’) & (~data[‘proxy_1’].isna()) & (~data[‘proxy_2’].isna())]

# Set the figure and subplot
fig, ax1 = plt.subplots(figsize=(4, 12))

# Fit GAM models
model_proxy_1 = LinearGAM(s(0, n_splines=20, lam=0.6)).fit(data_filtered[‘Depth’], data_filtered[‘proxy_1’])
model_proxy_2 = LinearGAM(s(0, n_splines=20, lam=0.6)).fit(data_filtered[‘Depth’], data_filtered[‘proxy_2’])

# Generate Depth values for plotting the models
depth_values = np.linspace(min(data_filtered[‘Depth’]), max(data_filtered[‘Depth’]), num=100)

# Predict values using the GAM models
predictions_proxy_1 = model_proxy_1.predict(depth_values)
predictions_proxy_2 = model_proxy_2.predict(depth_values)

# Set the style
plt.style.use(‘ggplot’)

# Add secondary x-axis for CA_Hop variable
ax2 = ax1.twiny()
ax2.plot(data_filtered[‘proxy_2’], data_filtered[‘Depth’], ‘o’, markersize=3, color=”#FF0000″)
ax2.plot(predictions_proxy_2, depth_values, ‘–‘, linewidth=0.75, color=”#FF0000″)
ax2.set_xlabel(r’Proxy 2′, color=”#FF0000″)
ax2.tick_params(axis=’x’, labelcolor=”#FF0000″)
ax2.set_xlim(25, 75)

# Plot results for d15N
ax1.plot(data_filtered[‘proxy_1’], data_filtered[‘Depth’], ‘o’, markersize=3, color=”#30638E”)
ax1.plot(predictions_proxy_1, depth_values, ‘-‘, linewidth=0.75, color=”#30638E”)
ax1.set_ylabel(“Depth (m)”)
ax1.set_xlabel(r’Proxy 1′, color=”#30638E”)
ax1.tick_params(axis=’x’, labelcolor=”#30638E”)
ax1.set_xlim(8.5, 4.5)

# Generate confidence intervals for the GAM models
confidence_intervals_proxy_2 = model_proxy_2.confidence_intervals(depth_values, width=0.95)
confidence_intervals_proxy_1 = model_proxy_1.confidence_intervals(depth_values, width=0.95)

# Plot confidence intervals
ax2.fill_betweenx(depth_values, confidence_intervals_proxy_2[:, 0], confidence_intervals_proxy_2[:, 1], alpha=0.2, color=”#FF0000″)
ax1.fill_betweenx(depth_values, confidence_intervals_proxy_1[:, 0], confidence_intervals_proxy_1[:, 1], alpha=0.2, color=”#30638E”)

# Customize the plot
for ax in [ax1, ax2]:
ax.set_xlabel(r’Proxy 1′, color=”#30638E”)
ax.tick_params(axis=’x’, labelcolor=”#30638E”)
ax.set_xlim(8.5, 4.5)
ax.set_ylim(590, 260)

ax.set_facecolor(‘white’)
ax.spines[‘bottom’].set_color(‘black’)
ax.spines[‘left’].set_color(‘black’)
ax.spines[‘top’].set_color(‘black’)
ax.spines[‘right’].set_color(‘black’)

ax.xaxis.grid(color=’#EAEAEA’)
ax.yaxis.grid(color=’#EAEAEA’)
ax.grid(True, which=’minor’, color=’#EAEAEA’, linestyle=’–‘)

# Set secondary x-axis limits
ax2.set_xlabel(r’Proxy 2′, color=”#FF0000″)
ax2.tick_params(axis=’x’, labelcolor=”#FF0000″)
ax2.set_xlim(25, 75)

plt.show()

# Save figures
fig.savefig(“Geochemical example.png”, dpi=300, bbox_inches=’tight’, format=’png’, width=6, height=2)
Geochemical data can be found below, apologies the heading have shifted to the left.

Sample_ID Drillcore_ID Depth proxy_1 proxy_2 proxy_3
Sample 1 drillcore example 271.95 NaN NaN 0.866
Sample 2 drillcore example 275.76 NaN NaN 0.786
Sample 3 drillcore example 279.35 NaN NaN NaN
Sample 4 drillcore example 289.85 NaN NaN 0.394
Sample 5 drillcore example 295.3 NaN NaN 0.745
Sample 6 drillcore example 313.43 5.5 59.5 1.429
Sample 7 drillcore example 330.4 NaN NaN NaN
Sample 8 drillcore example 338.8 6 73.3 0.926
Sample 9 drillcore example 341.1 5.8 50 1.208
Sample 10 drillcore example 365.2 6.6 72.4 1.844
Sample 11 drillcore example 371.76 7.5 71.3 0.799
Sample 12 drillcore example 376.4 6.8 56.7 1.354
Sample 13 drillcore example 382.22 NaN NaN 1.261
Sample 14 drillcore example 393.7 10.9 23.5 0.223
Sample 15 drillcore example 402.4 6.5 59.4 0.303
Sample 16 drillcore example 406.7 6.2 74.2 0.401
Sample 17 drillcore example 414.75 6.5 79.4 0.408
Sample 18 drillcore example 423.1 NaN NaN 0.497
Sample 19 drillcore example 429.7 6.7 81.5 0.445
Sample 20 drillcore example 443.7 7.1 62.3 0.657
Sample 21 drillcore example 450 7.1 75.8 0.758
Sample 22 drillcore example 452.1 6.2 56.1 0.783
Sample 23 drillcore example 457.2 6.6 68.9 0.631
Sample 24 drillcore example 461.83 7.8 65.7 0.802
Sample 25 drillcore example 465.3 5.9 47.6 0.74
Sample 26 drillcore example 470.7 6.3 69.8 0.878
Sample 27 drillcore example 474.4 6.6 60.6 0.927
Sample 28 drillcore example 477 7.1 64.8 0.826
Sample 29 drillcore example 478.2 5.9 53.4 0.751
Sample 30 drillcore example 485 6.5 71.2 0.971
Sample 31 drillcore example 489 6.4 72.1 0.948
Sample 32 drillcore example 492.98 NaN NaN 1.049
Sample 33 drillcore example 494.5 5.6 49.1 0.925
Sample 34 drillcore example 494.6 6.5 66 0.86
Sample 35 drillcore example 498.88 5.6 63.9 1.075
Sample 36 drillcore example 506 5.1 70.5 1.201
Sample 37 drillcore example 515.35 5.4 68.7 1.087
Sample 38 drillcore example 522.68 5.4 85.5 0.995
Sample 39 drillcore example 527.32 5.5 63 0.892
Sample 40 drillcore example 542.61 6.2 69.4 0.801
Sample 41 drillcore example 552.62 6.2 37.3 0.625
Sample 42 drillcore example 558.9 8 41.4 1.369
Sample 43 drillcore example 561.5 6 33.8 0.465
Sample 44 drillcore example 577 6.6 56.7 4.55
Sample 45 drillcore example 578.5 7.4 56.5 2.707