1.0 Measuring the Link Between Figure 8 Knots and Ascher Sum Summand Ranges

We are now equipped with two dataframes:

A Sums-and-Eight-Knots Dataframe showing Sum Information for pendant ascher sums, augmented with information about figure-eight-knot cords in the summand ranges.
An Eight-Knot-Sum-Summary “contingency table” which summarizes the above table by khipu.

Code

```{python}
import os
import numpy as np
from scipy import stats
import pandas as pd
from pandas import Series, DataFrame
from collections import Counter

import qollqa_chuspa as qc  # A Khipu Maker is known (in Quechua) as a Khipu Kamayuq
import utils_loom as uloom
import utils_khipu as ukhipu
import utils_pandas as upanda
import utils_kfg_locations as uloc
    
# Plotly
import plotly
from plotly.offline import iplot, init_notebook_mode
import plotly.graph_objects as go
import plotly.express as px
import plotly.figure_factory as ff
plotly.offline.init_notebook_mode(connected = False)
# Load all khipus
(khipu_dict, all_khipus) = qc.fetch_khipus()
```

1.1 Examining the Distribution of Figure-8-Knots vs Pendant Ascher Sums

First let’s look at two distributions:

A Khipu’s Pendant Ascher sums vs # Khipu’sFigure8 Knot Cords
A Khipu’s Pendant Ascher sums vs # Khipu’s Pendant Ascher Sums marked by Figure8 Knot Cords

As seen below the first distribution show’s a fairly wide standard deviation. However the second distribution brings the relation into a little more focus:

Code

```{python}
pendant_ascher_sums_df = pd.read_csv(f"{uloc.fieldmarks_data_dir()}/figure8knotsums.csv")
left_handed_eightknotsums_df = pd.read_csv(f"{uloc.fieldmarks_data_dir()}/left_handed_sums_8knot_types.csv")
right_handed_eightknotsums_df = pd.read_csv(f"{uloc.fieldmarks_data_dir()}/right_handed_sums_8knot_types.csv")

all_eightknotsums_df  = pd.read_csv(f"{uloc.fieldmarks_data_dir()}/all_sums_8knot_types.csv")
all_eightknotsums_df['num_total_cords'] = [khipu_dict[kfg_name].num_all_cords(include_subsidiaries=True) for kfg_name in all_eightknotsums_df['kfg_name'].to_list()]
all_eightknotsums_df['num_pendant_cords'] = [khipu_dict[kfg_name].num_pendant_cords() for kfg_name in all_eightknotsums_df['kfg_name'].to_list()]
```

Code

```{python}
fig = (px.scatter(all_eightknotsums_df, 
                    x='num_ascher_sums',
                    y='total_8knot_cords', 
                    color='num_pendant_cords', 
                    color_continuous_scale='Bluered_r', 
                    log_x=True, log_y=True,
                    title="By Khipu - # All Pendant Sum Cords vs # All Figure8Knot Cords - Hover over Dot for Detail",
                    labels={'num_ascher_sums':'# Basic Ascher Sums (Log scale)', 
                            'total_8knot_cords':'# Total Figure-8-Knot Cords (Log scale)',
                            'num_pendant_cords': '# Pendant Cords'
                            },
                    width=944, height=944,
                    trendline="ols",
                    hover_data=['kfg_name', 'num_ascher_sums', 'total_sum8knots', 'num_pendant_cords'])).show()
```

Code

```{python}
fig = (px.scatter(all_eightknotsums_df, 
                    x='num_ascher_sums',
                    y='total_sum8knots', 
                    color='num_pendant_cords', 
                    color_continuous_scale='Bluered_r', 
                    log_x=True, log_y=True,
                    title="By Khipu - # All Pendant Sum Cords vs # Figure-8-Knot Sum Marker Cords - Hover over Dot for Detail",
                    labels={'num_ascher_sums':'# Basic Ascher Sums (Log scale)', 
                            'total_sum8knots':'# Basic Ascher Sums with Figure 8 Knot Indicator (Log scale)', 
                            'num_pendant_cords': '# Pendant Cords'
                            },
                    width=944, height=944,
                    trendline="ols",
                    hover_data=['kfg_name', 'num_ascher_sums', 'total_sum8knots', 'num_pendant_cords'])).show()
```

1.2 Examining the Figure-8-Knot Cords over Summand Ranges

Similar to the Group/Cluster analysis that we did on Figure8Knot Cord Locations, we can compute Normalized Cord Locations for Figure-8-Knot Cords over Summand Ranges. In groups, the cords are contiguous. While this also true for pendant-pendant-sum summand ranges, it is not true for indexed-pendant-sums (ie. sum=p12+p22+p32+…) or colored-pendant-sums. Consequently, to generalize for normalized cord locations, we will use the index of the cord in its summand cords as our measurement metric, rather than the actual pendant index.

Code

```{python}
pendant_ascher_sums_df.head()
```

	kfg_name	cord_name	cord_index	cord_value	cord_color	num_summands	handedness	has_figure8knot_indicator	has_left_exact_8knot_cord	has_right_exact_8knot_cord	...	summand_string	sum_type	left_exact_pendant_8knot_cord_type	right_exact_pendant_8knot_cord_type	left_exact_subsidiary_8knot_cord_type	right_exact_subsidiary_8knot_cord_type	left_close_pendant_8knot_cord_type	right_close_pendant_8knot_cord_type	left_close_subsidiary_8knot_cord_type	right_close_subsidiary_8knot_cord_type
0	AK001	p24	[2, 0]	11	W	3	-2	True	False	False	...	W:YB:G@[1, 5]:4 + AB@[1, 6]:4 + AB@[1, 7]:3	pendant_pendant_sum	NaN	NaN	NaN	NaN	NaN	trailing_8knot	NaN	NaN
1	AK001	p33	[4, 0]	68	W	14	-8	True	True	False	...	W:YB:G@[1, 2]:1 + W:YB:G@[1, 3]:9 + W:YB:G@[1,...	pendant_pendant_sum	sole_8knot	NaN	NaN	NaN	NaN	NaN	NaN	NaN
2	AK001	p34	[4, 1]	81	W	14	-15	False	False	False	...	W:YB@[0, 12]:6 + W:YB@[0, 13]:6 + W:YB@[0, 14]...	pendant_pendant_sum	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN
3	AK001	p35	[4, 2]	33	YB	7	-7	True	False	False	...	W@[2, 1]:9 + W@[2, 2]:7 + W@[2, 3]:2 + W@[2, 4...	pendant_pendant_sum	NaN	NaN	NaN	NaN	trailing_8knot	NaN	NaN	NaN
4	AK001	p37	[4, 4]	27	W:YB:G	3	-12	True	True	False	...	W@[2, 0]:11 + W@[2, 1]:9 + W@[2, 2]:7	pendant_pendant_sum	trailing_8knot	NaN	NaN	NaN	NaN	NaN	NaN	NaN

5 rows × 31 columns

Code

```{python}
def in_left_half(test_index, num_items, include_middle=True):
    is_left=False
    if num_items == 0: is_left = False
    elif num_items == 1: is_left = include_middle
    else: is_left = test_index < float(num_items)/2.0 if include_middle else float(num_items-1)/2.0 
    return is_left

def in_right_half(test_index, num_items, include_middle=True):
    is_right=False
    if num_items == 0: is_right = False
    elif num_items == 1: is_right = include_middle
    elif test_index >= num_items: is_right = False    
    else: is_right = test_index >= float(num_items-1)/2.0 if include_middle else float(num_items)/2.0 
    return is_right

def leading_position(aCord, summands):
    return summands.index(aCord) 

def trailing_position(aCord, summands):
    return (len(summands) - summands.index(aCord)) -1
```

Code

```{python}
# Build a dictionary of khipu names, with associated summands (as list of (list of cords))
pendant_ascher_sums_kfg_names = ukhipu.lexigraphic_order(uloom.unique(pendant_ascher_sums_df['kfg_name'].to_list()))

summand_cord_dict = {kfg_name:[] for kfg_name in pendant_ascher_sums_kfg_names}
summand_8knot_cords = []

# Play with this number to reduce the influence of sums with only a few summands
# For example 5 still causes spikes at the end, but flattens the middle
# bandwith_limit_min = 2 # Default minimum

for _,sum_row in pendant_ascher_sums_df.iterrows():
    kfg_name = sum_row['kfg_name']
    the_khipu = khipu_dict[kfg_name]
    summand_cords = the_khipu.cords_from_sum_string(sum_row['summand_string'], one_based=False)
    # if len(summand_cords) <= bandwith_limit_min: 
    #    continue
    summand_cord_dict[kfg_name].append(summand_cords)
    for aCord in summand_cords:
        num_summands = len(summand_cords)
        the_leading_position = leading_position(aCord, summand_cords)
        the_trailing_position = trailing_position(aCord, summand_cords)
        cord_dict = {'kfg_name':kfg_name, 
                     'sum_cord':sum_row['cord_name'], 
                     'summand_cords': ",".join([cord.pendant_name for cord in summand_cords]),
                     'num_summands':num_summands, 
                     'cord_name':aCord.pendant_name, 
                     'is_subsidiary':aCord.has_eight_knot(include_pendant=False,include_subsidiaries=True), 
                     'cord':aCord,
                     'leading_position': leading_position(aCord, summand_cords),
                     'trailing_position': trailing_position(aCord, summand_cords),
                     'in_left_half': in_left_half(the_leading_position, num_summands, include_middle=False),
                     'in_right_half': in_right_half(the_trailing_position, num_summands, include_middle=False),
                     'normalized_position': float(summand_cords.index(aCord))/float(num_summands-1)
                     }
                     
        if aCord.has_eight_knot(include_pendants=True,include_subsidiaries=True):         
            summand_8knot_cords.append(cord_dict)

```

Code

```{python}
leading_figure_8_knot_cord_in_summand_ranges = [aCordDict['leading_position'] for aCordDict in summand_8knot_cords if aCordDict['in_left_half']] 
fig = (px.histogram(leading_figure_8_knot_cord_in_summand_ranges, nbins=160, width=944, height=500, 
                    title='Histogram of Left-Half Summand Range 8-Knot Cord Locations by LEADING Position from Left',
                    labels={'value':'Locations by LEADING Cord Position (from Left) in a Summand Range'},
                    )
        .update_layout(
            showlegend=False,
            font_family="Lucida Grande",
            font_color="black",
            font_size=12,
            title_font_family="Lucida Grande",
            title_font_color="black",
            legend_title_font_color="black")
        .show())  
```

Code

```{python}
trailing_figure_8_knot_cord_in_summand_ranges = [aCordDict['trailing_position'] for aCordDict in summand_8knot_cords if aCordDict['in_right_half']] 
fig = (px.histogram(trailing_figure_8_knot_cord_in_summand_ranges, nbins=160, width=944, height=500, 
                    title='Histogram of Right-Half Summand Range 8-Knot Cord Locations by Trailing Position from Right',
                    labels={'value':'Locations by Trailing Cord Position (from Right) in a Summand Range'},
                    )
        .update_layout(
            showlegend=False,
            font_family="Lucida Grande",
            font_color="black",
            font_size=12,
            title_font_family="Lucida Grande",
            title_font_color="black",
            legend_title_font_color="black")
        .show())
```

Code

```{python}
normalized_8knot_cord_in_summand_locs = [aCordDict['normalized_position'] for aCordDict in summand_8knot_cords if (aCordDict['in_left_half'] or aCordDict['in_right_half'])] 

fig = (px.histogram(normalized_8knot_cord_in_summand_locs, nbins=100, width=944, height=500, 
                    title='Histogram of Figure-8-Knot Normalized Cord Locations by Cord over a Summand Range')
        .update_layout(
            showlegend=False,
            font_family="Lucida Grande",
            font_color="black",
            font_size=12,
            title_font_family="Lucida Grande",
            title_font_color="black",
            legend_title_font_color="black")
        .show())
```

The spikes in the middle, at .333, .2, etc are the “harmonic” 1/n counts caused by 3 cord sequences, 4 cord sequences, etc., where the middle cord has an 8-knot (or its subsidiary), or the 2nd of 4 cords, etc..

2.0 Correlation

As a first step, let’s use the contigency table we built to evaluate which marks are most typical to identify summand ranges in an Ascher sum.

2.1 Left-Handed 8-Knot Summand Marker Correlation Coefficients

Using the R language/data-science tools, we can develop a crosstab of correlation coefficients:

Code

```{python}
""" This is R Code. Extracted from R environment

library(tidyverse)
library(corrplot)
library(Hmisc)

readRenviron("~/.Renviron")
k_LOOM_DIR <- Sys.getenv("LOOM_DIR")
k_FIELDMARKS_DATA_DIR <- paste(k_LOOM_DIR, "/data/FIELDMARKS/CSV", sep='')
setwd(k_FIELDMARKS_DATA_DIR)

plot_correlation_df <- function(aDF, aTitle, aCSVFilename) {
  precorrelation_df <-  select(aDF, -kfg_name) 
  
  the_cor_data  <-  cor(precorrelation_df, method="spearman")
  the_cor_data.tbl <- tibble(the_cor_data)
  write.csv(the_cor_data.tbl, file=aCSVFilename)
  
  corrplot(the_cor_data, method="ellipse", title=paste("\n\n",aTitle))
  }

plot_correlation_df(left_handed_eightknotsums_df, "Left-Handed Sums Marked by 8Knot Cords", "eightknotsums_left_handed_correlation.csv")
plot_correlation_df(right_handed_eightknotsums_df, "Right-Handed Sums Marked by 8Knot Cords", "eightknotsums_right_handed_correlation.csv")

""";
```

Code

```{python}
import tabulate
def clean_name(df):
    for old_column_name in df.columns:
        new_column_name = old_column_name.replace("the_cor_data.", "")  
        df = df.rename(columns={old_column_name: new_column_name})
    return df

def print_names(df):
    for old_column_name in df.columns:
        new_column_name = old_column_name.replace("_", " ").replace("num", "#").title()
        df = df.rename(columns={old_column_name: new_column_name})
    return df

def remove_totals(df):
    """ Remove the totals column from a correlation matrix """
    drop_columns = [column_name for column_name in df.columns if column_name.startswith("total")]
    for column_name in drop_columns:
        df = df.drop(column_name, axis=1)    
    
    # Drop total rows
    df = df.drop(df.index[4:12])
    return df

def remove_detail(df):
    """ Remove the details columns from a correlation matrix """
    drop_columns = [column_name for column_name in df.columns if column_name.startswith("left") or column_name.startswith("right")]
    for column_name in drop_columns:
        df = df.drop(column_name, axis=1)   
    
    df = df[0:len(df.columns)-1]
    
    return df

left_handed_8knotsums_correlation_df = pd.read_csv(f"{uloc.fieldmarks_data_dir()}/eightknotsums_left_handed_correlation.csv")
right_handed_8knotsums_correlation_df = pd.read_csv(f"{uloc.fieldmarks_data_dir()}/eightknotsums_right_handed_correlation.csv")

do_print_totals = False
if do_print_totals:
    totals_left_handed_8knotsums_correlation_df = remove_detail(clean_name(left_handed_8knotsums_correlation_df))
    totals_left_handed_8knotsums_correlation_df = totals_left_handed_8knotsums_correlation_df.rename(columns={'Unnamed: 0': 'markers'})
    totals_left_handed_8knotsums_correlation_df = print_names(totals_left_handed_8knotsums_correlation_df)
    totals_left_handed_8knotsums_correlation_df['Markers'] = totals_left_handed_8knotsums_correlation_df.columns[1:]

    totals_right_handed_8knotsums_correlation_df = remove_detail(clean_name(right_handed_8knotsums_correlation_df))
    totals_right_handed_8knotsums_correlation_df = totals_right_handed_8knotsums_correlation_df.rename(columns={'Unnamed: 0': 'markers'})
    totals_right_handed_8knotsums_correlation_df = print_names(totals_right_handed_8knotsums_correlation_df)
    totals_right_handed_8knotsums_correlation_df['Markers'] = totals_right_handed_8knotsums_correlation_df.columns[1:]

    print("\nLeft-Handed 8Knot Sum Totals Correlation:")
    print(  "=========================================\n")
    print(tabulate.tabulate(totals_left_handed_8knotsums_correlation_df, headers='keys', tablefmt='html')    )
    print("\nRight-Handed 8Knot Sum Totals Correlation:")
    print(  "==========================================\n")
    print(tabulate.tabulate(totals_right_handed_8knotsums_correlation_df, headers='keys', tablefmt='html')    )

do_print_details = False
if do_print_details:
    details_left_handed_8knotsums_correlation_df = remove_totals(clean_name(left_handed_8knotsums_correlation_df))
    details_left_handed_8knotsums_correlation_df = details_left_handed_8knotsums_correlation_df.rename(columns={'Unnamed: 0': 'markers'})
    details_left_handed_8knotsums_correlation_df = print_names(details_left_handed_8knotsums_correlation_df)
    details_left_handed_8knotsums_correlation_df['Markers'] = details_left_handed_8knotsums_correlation_df.columns[1:]


    details_right_handed_8knotsums_correlation_df = remove_totals(clean_name(right_handed_8knotsums_correlation_df))
    details_right_handed_8knotsums_correlation_df = details_right_handed_8knotsums_correlation_df.rename(columns={'Unnamed: 0': 'markers'})
    details_right_handed_8knotsums_correlation_df = print_names(details_right_handed_8knotsums_correlation_df)
    details_right_handed_8knotsums_correlation_df['Markers'] = details_right_handed_8knotsums_correlation_df.columns[1:]

    print("\nLeft-Handed 8Knot Sum Details Correlation:")
    print(  "==========================================\n")
    print(tabulate.tabulate(details_left_handed_8knotsums_correlation_df, headers='keys', tablefmt='html'))
    print("\nRight-Handed 8Knot Sum Details Correlation:")
    print(  "===========================================\n")
    print(tabulate.tabulate(details_right_handed_8knotsums_correlation_df, headers='keys', tablefmt='html'))  
```

2.1 Correlation for Left-Handed 8Knot Sums

Click on image to view larger

2.1.1 Totals Correlation Matrix for Left-Handed 8Knot Sums

	Markers	# Ascher Sums	# Pps Sums	# Cps Sums	# Ips Sums	Total 8Knot Cords	Total Sum8Knots	Total Sole8Knots	Total Trailing8Knots	Total Exact Sole8Knots	Total Exact Trailing8Knots	Total Close Sole8Knots	Total Close Trailing8Knots
0	# Ascher Sums	1	0.948807	0.742891	0.679007	0.570751	0.628623	0.505053	0.579329	0.485423	0.569114	0.503967	0.540172
1	# Pps Sums	0.948807	1	0.588574	0.593235	0.544039	0.594524	0.475439	0.588339	0.459165	0.564179	0.468157	0.558728
2	# Cps Sums	0.742891	0.588574	1	0.469379	0.532935	0.582093	0.483551	0.469604	0.458989	0.469438	0.47675	0.432925
3	# Ips Sums	0.679007	0.593235	0.469379	1	0.397665	0.431915	0.35567	0.388948	0.338436	0.384091	0.386655	0.371352
4	Total 8Knot Cords	0.570751	0.544039	0.532935	0.397665	1	0.861506	0.744322	0.649461	0.707029	0.62683	0.708696	0.577937
5	Total Sum8Knots	0.628623	0.594524	0.582093	0.431915	0.861506	1	0.900431	0.639022	0.835829	0.655497	0.857926	0.538485
6	Total Sole8Knots	0.505053	0.475439	0.483551	0.35567	0.744322	0.900431	1	0.399872	0.945118	0.389912	0.912097	0.361942
7	Total Trailing8Knots	0.579329	0.588339	0.469604	0.388948	0.649461	0.639022	0.399872	1	0.388338	0.903811	0.384371	0.893418
8	Total Exact Sole8Knots	0.485423	0.459165	0.458989	0.338436	0.707029	0.835829	0.945118	0.388338	1	0.391587	0.77947	0.339827
9	Total Exact Trailing8Knots	0.569114	0.564179	0.469438	0.384091	0.62683	0.655497	0.389912	0.903811	0.391587	1	0.360856	0.684955
10	Total Close Sole8Knots	0.503967	0.468157	0.47675	0.386655	0.708696	0.857926	0.912097	0.384371	0.77947	0.360856	1	0.362621
11	Total Close Trailing8Knots	0.540172	0.558728	0.432925	0.371352	0.577937	0.538485	0.361942	0.893418	0.339827	0.684955	0.362621	1

2.1.2 Details Correlation Matrix for Left-Handed 8Knot Sums

	Markers	# Ascher Sums	# Pps Sums	# Cps Sums	# Ips Sums	Left Exact Pendant 8Knot Cord Type Sole8Knot	Left Exact Pendant 8Knot Cord Type Trailing8Knot	Right Exact Pendant 8Knot Cord Type Sole8Knot	Right Exact Pendant 8Knot Cord Type Trailing8Knot	Left Exact Subsidiary 8Knot Cord Type Sole8Knot	Left Exact Subsidiary 8Knot Cord Type Trailing8Knot	Right Exact Subsidiary 8Knot Cord Type Sole8Knot	Right Exact Subsidiary 8Knot Cord Type Trailing8Knot	Left Close Pendant 8Knot Cord Type Sole8Knot	Left Close Pendant 8Knot Cord Type Trailing8Knot	Right Close Pendant 8Knot Cord Type Sole8Knot	Right Close Pendant 8Knot Cord Type Trailing8Knot	Left Close Subsidiary 8Knot Cord Type Sole8Knot	Left Close Subsidiary 8Knot Cord Type Trailing8Knot	Right Close Subsidiary 8Knot Cord Type Sole8Knot	Right Close Subsidiary 8Knot Cord Type Trailing8Knot
0	# Ascher Sums	1	0.948807	0.742891	0.679007	0.41734	0.513025	0.427799	0.545608	0.363461	0.238475	0.395055	0.242144	0.378807	0.413304	0.369599	0.546995	0.41413	0.248631	0.369947	0.210695
1	# Pps Sums	0.948807	1	0.588574	0.593235	0.38125	0.517785	0.382904	0.529056	0.351608	0.245243	0.389071	0.239335	0.336569	0.431289	0.309897	0.563085	0.407803	0.231669	0.353896	0.217777
2	# Cps Sums	0.742891	0.588574	1	0.469379	0.416983	0.411542	0.46799	0.471311	0.349467	0.184721	0.337134	0.179364	0.391751	0.330903	0.430067	0.446049	0.342143	0.204782	0.319785	0.148004
3	# Ips Sums	0.679007	0.593235	0.469379	1	0.284562	0.362618	0.302116	0.39742	0.225712	0.132278	0.271894	0.172722	0.327295	0.270626	0.284538	0.385501	0.326215	0.181261	0.26559	0.171859
12	Left Exact Pendant 8Knot Cord Type Sole8Knot	0.41734	0.38125	0.416983	0.284562	1	0.214084	0.688097	0.246541	0.375901	0.0457201	0.360164	0.0878349	0.703917	0.182259	0.502263	0.208521	0.303417	0.0572445	0.308434	0.0547797
13	Left Exact Pendant 8Knot Cord Type Trailing8Knot	0.513025	0.517785	0.411542	0.362618	0.214084	1	0.245139	0.639403	0.279633	0.24995	0.342603	0.254849	0.20946	0.53409	0.145712	0.538393	0.270505	0.30012	0.301544	0.222699
14	Right Exact Pendant 8Knot Cord Type Sole8Knot	0.427799	0.382904	0.46799	0.302116	0.688097	0.245139	1	0.237838	0.36745	0.0967522	0.331345	0.100344	0.603116	0.195804	0.652987	0.247054	0.342444	0.116278	0.315668	0.0933847
15	Right Exact Pendant 8Knot Cord Type Trailing8Knot	0.545608	0.529056	0.471311	0.39742	0.246541	0.639403	0.237838	1	0.283334	0.22287	0.385689	0.259653	0.221541	0.446826	0.13447	0.595174	0.333079	0.292786	0.261043	0.27072
16	Left Exact Subsidiary 8Knot Cord Type Sole8Knot	0.363461	0.351608	0.349467	0.225712	0.375901	0.279633	0.36745	0.283334	1	0.268888	0.685392	0.281259	0.326636	0.177943	0.301021	0.257645	0.647528	0.168586	0.62934	0.296692
17	Left Exact Subsidiary 8Knot Cord Type Trailing8Knot	0.238475	0.245243	0.184721	0.132278	0.0457201	0.24995	0.0967522	0.22287	0.268888	1	0.15891	0.552854	0.0136631	0.234946	0.0178539	0.25461	0.190897	0.367369	0.157244	0.527578
18	Right Exact Subsidiary 8Knot Cord Type Sole8Knot	0.395055	0.389071	0.337134	0.271894	0.360164	0.342603	0.331345	0.385689	0.685392	0.15891	1	0.292359	0.356131	0.249186	0.272893	0.332134	0.59031	0.24276	0.695243	0.29347
19	Right Exact Subsidiary 8Knot Cord Type Trailing8Knot	0.242144	0.239335	0.179364	0.172722	0.0878349	0.254849	0.100344	0.259653	0.281259	0.552854	0.292359	1	0.0929279	0.208627	0.0452505	0.271329	0.227464	0.482741	0.23339	0.61148
20	Left Close Pendant 8Knot Cord Type Sole8Knot	0.378807	0.336569	0.391751	0.327295	0.703917	0.20946	0.603116	0.221541	0.326636	0.0136631	0.356131	0.0929279	1	0.126613	0.575509	0.170884	0.338788	0.12239	0.33364	0.0611192
21	Left Close Pendant 8Knot Cord Type Trailing8Knot	0.413304	0.431289	0.330903	0.270626	0.182259	0.53409	0.195804	0.446826	0.177943	0.234946	0.249186	0.208627	0.126613	1	0.122714	0.490765	0.251387	0.199824	0.274054	0.252267
22	Right Close Pendant 8Knot Cord Type Sole8Knot	0.369599	0.309897	0.430067	0.284538	0.502263	0.145712	0.652987	0.13447	0.301021	0.0178539	0.272893	0.0452505	0.575509	0.122714	1	0.138937	0.297894	0.0786808	0.275281	0.0889176
23	Right Close Pendant 8Knot Cord Type Trailing8Knot	0.546995	0.563085	0.446049	0.385501	0.208521	0.538393	0.247054	0.595174	0.257645	0.25461	0.332134	0.271329	0.170884	0.490765	0.138937	1	0.357727	0.328861	0.319765	0.206607
24	Left Close Subsidiary 8Knot Cord Type Sole8Knot	0.41413	0.407803	0.342143	0.326215	0.303417	0.270505	0.342444	0.333079	0.647528	0.190897	0.59031	0.227464	0.338788	0.251387	0.297894	0.357727	1	0.228697	0.59576	0.312794
25	Left Close Subsidiary 8Knot Cord Type Trailing8Knot	0.248631	0.231669	0.204782	0.181261	0.0572445	0.30012	0.116278	0.292786	0.168586	0.367369	0.24276	0.482741	0.12239	0.199824	0.0786808	0.328861	0.228697	1	0.182084	0.385694
26	Right Close Subsidiary 8Knot Cord Type Sole8Knot	0.369947	0.353896	0.319785	0.26559	0.308434	0.301544	0.315668	0.261043	0.62934	0.157244	0.695243	0.23339	0.33364	0.274054	0.275281	0.319765	0.59576	0.182084	1	0.271141
27	Right Close Subsidiary 8Knot Cord Type Trailing8Knot	0.210695	0.217777	0.148004	0.171859	0.0547797	0.222699	0.0933847	0.27072	0.296692	0.527578	0.29347	0.61148	0.0611192	0.252267	0.0889176	0.206607	0.312794	0.385694	0.271141	1

2.2 Right-Handed 8-Knot Summand Marker Correlation Coefficients

Click on image to view larger

2.2.1 Totals Correlation Matrix for Right-Handed 8Knot Sums

	Markers	# Ascher Sums	# Pps Sums	# Cps Sums	# Ips Sums	Total 8Knot Cords	Total Sum8Knots	Total Sole8Knots	Total Trailing8Knots	Total Exact Sole8Knots	Total Exact Trailing8Knots	Total Close Sole8Knots	Total Close Trailing8Knots
0	# Ascher Sums	1	0.939332	0.800366	0.713307	0.573015	0.664177	0.509671	0.621999	0.532267	0.632791	0.507827	0.568578
1	# Pps Sums	0.939332	1	0.655455	0.598529	0.53477	0.620943	0.477433	0.605176	0.500382	0.610611	0.47216	0.561434
2	# Cps Sums	0.800366	0.655455	1	0.527032	0.549278	0.59831	0.475298	0.536619	0.493177	0.539519	0.472877	0.491748
3	# Ips Sums	0.713307	0.598529	0.527032	1	0.436404	0.492266	0.355097	0.495715	0.366676	0.503991	0.369016	0.447228
4	Total 8Knot Cords	0.573015	0.53477	0.549278	0.436404	1	0.880645	0.763182	0.637174	0.760499	0.624552	0.727233	0.574227
5	Total Sum8Knots	0.664177	0.620943	0.59831	0.492266	0.880645	1	0.875236	0.65349	0.846803	0.685631	0.854764	0.551337
6	Total Sole8Knots	0.509671	0.477433	0.475298	0.355097	0.763182	0.875236	1	0.372837	0.968696	0.373629	0.943621	0.332265
7	Total Trailing8Knots	0.621999	0.605176	0.536619	0.495715	0.637174	0.65349	0.372837	1	0.38226	0.912383	0.375496	0.904691
8	Total Exact Sole8Knots	0.532267	0.500382	0.493177	0.366676	0.760499	0.846803	0.968696	0.38226	1	0.39136	0.867862	0.338447
9	Total Exact Trailing8Knots	0.632791	0.610611	0.539519	0.503991	0.624552	0.685631	0.373629	0.912383	0.39136	1	0.377167	0.727431
10	Total Close Sole8Knots	0.507827	0.47216	0.472877	0.369016	0.727233	0.854764	0.943621	0.375496	0.867862	0.377167	1	0.328479
11	Total Close Trailing8Knots	0.568578	0.561434	0.491748	0.447228	0.574227	0.551337	0.332265	0.904691	0.338447	0.727431	0.328479	1

2.2.2 Details Correlation Matrix for Right-Handed 8Knot Sums

	Markers	# Ascher Sums	# Pps Sums	# Cps Sums	# Ips Sums	Left Exact Pendant 8Knot Cord Type Sole8Knot	Left Exact Pendant 8Knot Cord Type Trailing8Knot	Right Exact Pendant 8Knot Cord Type Sole8Knot	Right Exact Pendant 8Knot Cord Type Trailing8Knot	Left Exact Subsidiary 8Knot Cord Type Sole8Knot	Left Exact Subsidiary 8Knot Cord Type Trailing8Knot	Right Exact Subsidiary 8Knot Cord Type Sole8Knot	Right Exact Subsidiary 8Knot Cord Type Trailing8Knot	Left Close Pendant 8Knot Cord Type Sole8Knot	Left Close Pendant 8Knot Cord Type Trailing8Knot	Right Close Pendant 8Knot Cord Type Sole8Knot	Right Close Pendant 8Knot Cord Type Trailing8Knot	Left Close Subsidiary 8Knot Cord Type Sole8Knot	Left Close Subsidiary 8Knot Cord Type Trailing8Knot	Right Close Subsidiary 8Knot Cord Type Sole8Knot	Right Close Subsidiary 8Knot Cord Type Trailing8Knot
0	# Ascher Sums	1	0.939332	0.800366	0.713307	0.399687	0.604616	0.44337	0.591881	0.372098	0.236934	0.397923	0.272567	0.387249	0.539967	0.395177	0.523485	0.398641	0.306553	0.408651	0.235105
1	# Pps Sums	0.939332	1	0.655455	0.598529	0.402435	0.583419	0.422585	0.586111	0.329716	0.229645	0.355146	0.263212	0.375652	0.533139	0.361325	0.524896	0.386428	0.286542	0.359598	0.219833
2	# Cps Sums	0.800366	0.655455	1	0.527032	0.37661	0.522281	0.423763	0.510917	0.356355	0.197998	0.395577	0.240902	0.380441	0.47233	0.405324	0.445796	0.364102	0.275713	0.413595	0.220166
3	# Ips Sums	0.713307	0.598529	0.527032	1	0.249146	0.490033	0.337277	0.462593	0.309	0.218203	0.279618	0.195836	0.300178	0.394717	0.289449	0.42989	0.288791	0.260018	0.312279	0.195163
12	Left Exact Pendant 8Knot Cord Type Sole8Knot	0.399687	0.402435	0.37661	0.249146	1	0.178685	0.648193	0.209645	0.283104	0.0124537	0.337608	-0.0148578	0.738293	0.195852	0.610128	0.15968	0.370028	0.012975	0.314333	0.0148751
13	Left Exact Pendant 8Knot Cord Type Trailing8Knot	0.604616	0.583419	0.522281	0.490033	0.178685	1	0.271785	0.697343	0.287746	0.228418	0.283734	0.312874	0.216206	0.668737	0.210201	0.577349	0.287554	0.328771	0.275379	0.277126
14	Right Exact Pendant 8Knot Cord Type Sole8Knot	0.44337	0.422585	0.423763	0.337277	0.648193	0.271785	1	0.271077	0.365724	0.0592264	0.381023	0.0505534	0.679436	0.240034	0.715463	0.191983	0.430866	0.135784	0.396913	0.082391
15	Right Exact Pendant 8Knot Cord Type Trailing8Knot	0.591881	0.586111	0.510917	0.462593	0.209645	0.697343	0.271077	1	0.317658	0.24441	0.302013	0.357137	0.232217	0.602267	0.181283	0.638875	0.314117	0.379622	0.325728	0.308684
16	Left Exact Subsidiary 8Knot Cord Type Sole8Knot	0.372098	0.329716	0.356355	0.309	0.283104	0.287746	0.365724	0.317658	1	0.310723	0.712757	0.269956	0.314619	0.253967	0.361568	0.190594	0.692194	0.304425	0.702125	0.285374
17	Left Exact Subsidiary 8Knot Cord Type Trailing8Knot	0.236934	0.229645	0.197998	0.218203	0.0124537	0.228418	0.0592264	0.24441	0.310723	1	0.303167	0.664067	-0.0186791	0.259496	0.00223261	0.248021	0.308222	0.588608	0.276163	0.583068
18	Right Exact Subsidiary 8Knot Cord Type Sole8Knot	0.397923	0.355146	0.395577	0.279618	0.337608	0.283734	0.381023	0.302013	0.712757	0.303167	1	0.257631	0.36188	0.265124	0.372944	0.249857	0.698246	0.296793	0.714776	0.20677
19	Right Exact Subsidiary 8Knot Cord Type Trailing8Knot	0.272567	0.263212	0.240902	0.195836	-0.0148578	0.312874	0.0505534	0.357137	0.269956	0.664067	0.257631	1	0.0312377	0.322955	0.0425252	0.33746	0.252457	0.685554	0.260263	0.694134
20	Left Close Pendant 8Knot Cord Type Sole8Knot	0.387249	0.375652	0.380441	0.300178	0.738293	0.216206	0.679436	0.232217	0.314619	-0.0186791	0.36188	0.0312377	1	0.149124	0.643274	0.156699	0.341948	0.034155	0.329152	0.0292368
21	Left Close Pendant 8Knot Cord Type Trailing8Knot	0.539967	0.533139	0.47233	0.394717	0.195852	0.668737	0.240034	0.602267	0.253967	0.259496	0.265124	0.322955	0.149124	1	0.192524	0.589833	0.238974	0.350788	0.309794	0.263385
22	Right Close Pendant 8Knot Cord Type Sole8Knot	0.395177	0.361325	0.405324	0.289449	0.610128	0.210201	0.715463	0.181283	0.361568	0.00223261	0.372944	0.0425252	0.643274	0.192524	1	0.132001	0.362932	0.0687638	0.372978	0.0527134
23	Right Close Pendant 8Knot Cord Type Trailing8Knot	0.523485	0.524896	0.445796	0.42989	0.15968	0.577349	0.191983	0.638875	0.190594	0.248021	0.249857	0.33746	0.156699	0.589833	0.132001	1	0.238546	0.361342	0.311055	0.261411
24	Left Close Subsidiary 8Knot Cord Type Sole8Knot	0.398641	0.386428	0.364102	0.288791	0.370028	0.287554	0.430866	0.314117	0.692194	0.308222	0.698246	0.252457	0.341948	0.238974	0.362932	0.238546	1	0.322512	0.734032	0.273338
25	Left Close Subsidiary 8Knot Cord Type Trailing8Knot	0.306553	0.286542	0.275713	0.260018	0.012975	0.328771	0.135784	0.379622	0.304425	0.588608	0.296793	0.685554	0.034155	0.350788	0.0687638	0.361342	0.322512	1	0.292727	0.715569
26	Right Close Subsidiary 8Knot Cord Type Sole8Knot	0.408651	0.359598	0.413595	0.312279	0.314333	0.275379	0.396913	0.325728	0.702125	0.276163	0.714776	0.260263	0.329152	0.309794	0.372978	0.311055	0.734032	0.292727	1	0.249226
27	Right Close Subsidiary 8Knot Cord Type Trailing8Knot	0.235105	0.219833	0.220166	0.195163	0.0148751	0.277126	0.082391	0.308684	0.285374	0.583068	0.20677	0.694134	0.0292368	0.263385	0.0527134	0.261411	0.273338	0.715569	0.249226	1

3.0 Chi-squared test between Basic Ascher Sums and Figure 8 Knot Cords

Code

```{python}
from scipy.stats import chi2_contingency

def perform_chi_squared_8knot_test(df):
    chi_sq_input_df = df[['num_ascher_sums', 'total_sum8knots', 'total_sole8knots', 'total_trailing8knots']]
    chi_sq_input_df = chi_sq_input_df[chi_sq_input_df['num_ascher_sums'] > 0]
    
    # Perform chi-squared test
    chi2, p, dof, expected = chi2_contingency(chi_sq_input_df)

    # Print the results
    print(f"p-value: {p:e}")
    return chi_sq_input_df

print("\nChi-squared  p values for Left-Handed 8Knot Sums")
print(  "================================================")
left_chi_sq_df = perform_chi_squared_8knot_test(left_handed_eightknotsums_df)

print("\nChi-squared  p values for Right-Handed 8Knot Sums")
print(  "=================================================")
right_chi_sq_df = perform_chi_squared_8knot_test(right_handed_eightknotsums_df)
```


Chi-squared  p values for Left-Handed 8Knot Sums
================================================
p-value: 0.000000e+00

Chi-squared  p values for Right-Handed 8Knot Sums
=================================================
p-value: 0.000000e+00

Code

```{python}
print("Left-Handed Chi-Squared Input Data:")
left_chi_sq_df.head(5)
```

Left-Handed Chi-Squared Input Data:

	num_ascher_sums	total_sum8knots	total_sole8knots	total_trailing8knots
0	6	3	2	3
1	1	1	1	1
2	1	0	0	0
3	4	10	7	4
4	6	4	3	2

Code

```{python}
print("Right-Handed Chi-Squared Input Data:")
right_chi_sq_df.head(5)
```

Right-Handed Chi-Squared Input Data:

	num_ascher_sums	total_sum8knots	total_sole8knots	total_trailing8knots
0	1	0	0	0
1	5	0	0	0
2	2	7	5	2
3	3	9	5	1
4	70	97	59	7

4.0 Angle Between Pendant Ascher Sums vs. Figure-8-Knot Markers

What is the vector angle between pendant Ascher sums, and the various figure-eight-knot markers (i.e. how closely do they match?)

Code

```{python}
print(f"Angle between num_pendant_ascher_sums and:")
for i in range(2, len(all_eightknotsums_df.columns)):
    print(f"\t{all_eightknotsums_df.columns[i]}: {uloom.degrees_between(all_eightknotsums_df['num_ascher_sums'].to_list(), all_eightknotsums_df[all_eightknotsums_df.columns[i]].to_list()):.1f}°")

```

Angle between num_pendant_ascher_sums and:
    num_pps_sums: 18.1°
    num_cps_sums: 23.5°
    num_ips_sums: 31.1°
    total_8knot_cords: 49.3°
    total_sum8knots: 44.5°
    total_sole8knots: 49.1°
    total_trailing8knots: 42.1°
    total_exact_sole8knots: 45.4°
    total_exact_trailing8knots: 39.6°
    total_close_sole8knots: 54.5°
    total_close_trailing8knots: 48.1°
    left_exact_pendant_8knot_cord_type_sole8knot: 45.8°
    left_exact_pendant_8knot_cord_type_trailing8knot: 42.5°
    right_exact_pendant_8knot_cord_type_sole8knot: 48.4°
    right_exact_pendant_8knot_cord_type_trailing8knot: 39.4°
    left_exact_subsidiary_8knot_cord_type_sole8knot: 64.6°
    left_exact_subsidiary_8knot_cord_type_trailing8knot: 65.1°
    right_exact_subsidiary_8knot_cord_type_sole8knot: 61.2°
    right_exact_subsidiary_8knot_cord_type_trailing8knot: 67.4°
    left_close_pendant_8knot_cord_type_sole8knot: 57.1°
    left_close_pendant_8knot_cord_type_trailing8knot: 49.4°
    right_close_pendant_8knot_cord_type_sole8knot: 59.7°
    right_close_pendant_8knot_cord_type_trailing8knot: 49.6°
    left_close_subsidiary_8knot_cord_type_sole8knot: 63.6°
    left_close_subsidiary_8knot_cord_type_trailing8knot: 70.8°
    right_close_subsidiary_8knot_cord_type_sole8knot: 62.6°
    right_close_subsidiary_8knot_cord_type_trailing8knot: 69.3°
    num_total_cords: 39.6°
    num_pendant_cords: 33.2°

5.0 Figure-8-Knot Cords Marking Sums vs Sums Marked by Figure-8-Knot Cords.

There is a difference between:

The number of 8 knot cords that mark an ascher sum and
The number of ascher sums marked by an eight knot cord

The reason is that we allow ascher sums to be marked by neighboring cords, as well as the left and right edges of the summand range. Consequently one cord can serve as a left-edge summand marker, and a close right-edge of a previous summand range. A similar duplicity exists for right/left summand ranges. For example, in KH0696, pendant p24 marks an exact sum relationship with p38, and a left close relationship with sum cord p35. In the figure8 knot dataframe we mark this as figure8knot p24 matches with p38, and leave it at that, ignoring p35. But in the Ascher sum dataframe we record both.

Code

```{python}
aDF=pendant_ascher_sums_df    
khipu_df = aDF[aDF['kfg_name']=='AK001'].fillna('')
khipu_df = khipu_df[khipu_df['has_figure8knot_indicator']==True]
print("Sums with a Figure 8 Knot Indicator for AK001")
khipu_df.head(10)
```

Sums with a Figure 8 Knot Indicator for AK001

	kfg_name	cord_name	cord_index	cord_value	cord_color	num_summands	handedness	has_figure8knot_indicator	has_left_exact_8knot_cord	has_right_exact_8knot_cord	...	summand_string	sum_type	left_exact_pendant_8knot_cord_type	left_close_pendant_8knot_cord_type	right_close_pendant_8knot_cord_type
0	AK001	p24	[2, 0]	11	W	3	-2	True	False	False	...	W:YB:G@[1, 5]:4 + AB@[1, 6]:4 + AB@[1, 7]:3	pendant_pendant_sum			trailing_8knot
1	AK001	p33	[4, 0]	68	W	14	-8	True	True	False	...	W:YB:G@[1, 2]:1 + W:YB:G@[1, 3]:9 + W:YB:G@[1,...	pendant_pendant_sum	sole_8knot
3	AK001	p35	[4, 2]	33	YB	7	-7	True	False	False	...	W@[2, 1]:9 + W@[2, 2]:7 + W@[2, 3]:2 + W@[2, 4...	pendant_pendant_sum		trailing_8knot
4	AK001	p37	[4, 4]	27	W:YB:G	3	-12	True	True	False	...	W@[2, 0]:11 + W@[2, 1]:9 + W@[2, 2]:7	pendant_pendant_sum	trailing_8knot
5	AK001	p38	[4, 5]	13	W:YB	3	-9	True	True	False	...	W@[2, 4]:1 + AB@[2, 5]:4 + AB@[2, 6]:8	pendant_pendant_sum	sole_8knot

5 rows × 31 columns

Code

```{python}
aKFGName = 'AK001'
from fieldmark_figure8knots import FieldmarkFigure8Knots
figure8knot_fieldmark = FieldmarkFigure8Knots()  
aDF = figure8knot_fieldmark.relations_df()
khipu_df = aDF[aDF['kfg_name']==aKFGName].fillna('')
khipu_df = khipu_df[khipu_df['sum_type'].str.len() > 0]

print("Figure 8 Knot Cords with Sum Indicators for AK001")
khipu_df.head(10)
```

Figure 8 Knot Cords with Sum Indicators for AK001

	kfg_name	cord_name	cord_value	cord_color	eight_knot_type	is_subsidiary	is_sum_match	sum_type	handedness	sum_cord_name	num_summands	is_edge_match	is_neighbor_match	match_side	is_parity_bit	summand_string
0	AK001	p18	1	W:YB:G	sole_8knot	False	True	pendant_pendant_sums	left	p33	14	True	False	left	True	p18 + p19 + p20 + p21 + p22 + p23 + p24 + p25 ...
1	AK001	p24	11	W	trailing_8knot	False	True	pendant_pendant_sums	left	p24	3	False	True	right	False	p21 + p22 + p23
2	AK001	p24	11	W	trailing_8knot	False	True	pendant_pendant_sums	left	p37	3	True	False	left	False	p24 + p25 + p26
3	AK001	p28	1	W	sole_8knot	False	True	pendant_pendant_sums	left	p38	3	True	False	left	True	p28 + p29 + p30

Appendix.

One question to ask is what if figure8knots occur over a khipu because of 1 being the last digit

We’ve already seen above that figure8knots are NOT uniformly distributed over a summand range, but instead bunched at the “edges”. If a khipu’s numbers are “randomly distributed” then there is a 1% chance of a sole 8 knot occuring every 100 cords. Clearly the frequency of sole 8 knot cords is not that:

Code

```{python}
from fieldmark_figure8knots import FieldmarkFigure8Knots
import utils_math as umath
figure8knot_fieldmark = FieldmarkFigure8Knots()
the_khipu_df = figure8knot_fieldmark.fieldmark_df()
eightknot_kfg_names = the_khipu_df['kfg_name'].to_list()
the_khipu_df.head()
```

	kfg_name	num_pendant_cords	num_subsidiary_cords	num_8knot_cords	num_pendant_8knot_cords	num_subsidiary_8knot_cords	num_sole_8knot_cords	num_pendant_sole_8_knot_cords	num_subsidiary_sole_8_knot_cords	num_trailing_8knot_cords	num_pendant_trailing_8_knot_cords	num_subsidiary_trailing_8_knot_cords	num_ascher_sums	num_8knot_sums	sole_8knot_cord_positions	trailing_8knot_cord_positions
0	AK001	38	1	3	3	0	2	2	0	1	1	0	7	4	p18, p28	p24
1	AK002	77	6	6	5	1	0	0	0	5	4	1	5	0	NaN	p2, p27, p29s1, p37, p42
2	AS001	14	17	6	2	4	4	0	4	2	2	0	0	0	p1s1, p2s2, p10s1, p12s2	p7, p13
3	AS002	7	8	2	0	2	2	0	2	0	0	0	0	0	p1s1, p3s1	NaN
4	AS003	7	2	0	0	0	0	0	0	0	0	0	0	0	NaN	NaN

Code

```{python}
num_pendant_cords = the_khipu_df['num_pendant_cords'].to_list()
num_sole_8knots = the_khipu_df['num_sole_8knot_cords'].to_list()
pct_sole_8knots = [round(100.0*float(num_sole_8knots[i])/float(num_pendant_cords[i]),1) for i in range(len(num_pendant_cords))]
pct_sole_8knots_df = pd.DataFrame({'kfg_name':eightknot_kfg_names, 'num_pendant_cords':num_pendant_cords, 'num_sole_8knots':num_sole_8knots, 'pct_sole_8knots':pct_sole_8knots}) 

import utils_math as umath
print("Statistics for % Sole 8 Knot Cords vs Pendant Cords")
print(umath.get_statistics(pct_sole_8knots))

fig = (px.scatter(pct_sole_8knots_df, 
                    x='num_pendant_cords',
                    y='num_sole_8knots', 
                    color='pct_sole_8knots', 
                    color_continuous_scale='Bluered_r', 
                    log_x=True, log_y=True,
                    title="By Khipu - # Pendant Cords vs # Sole 8 Knot Cords - Hover over Dot for Detail",
                    labels={'num_pendant_cords':'# Pendant Cords (Log scale)', 
                            'num_sole_8knots':'# Sole Figure-8-Knot Cords (Log scale)',
                            'pct_sole_8knots': '% Sole 8 Knot Cords vs Pendant Cords',
                            'kfg_name':'Khipu Name',
                            },
                    width=944, height=944,
                    trendline="ols",
                    hover_data=['num_pendant_cords', 'num_sole_8knots', 'pct_sole_8knots', 'kfg_name'])).show()

```

Statistics for % Sole 8 Knot Cords vs Pendant Cords
{'min': 0.0, 'mean': 14.603915662650602, 'median': 3.1500000000000004, 'max': 400.0, 'stddev': 31.583377881412815, 'range': 400.0, 'spread': 0.15791688940706408}

So clearly Sole8Knots are occuring at a much higher rate than 1% (14.6%!), with a trendline of 11%.

A similar line of reasoning can occur with trailing 8 knots. For example in 100 cords we have a 10% chance of getting cords that end with 1 (ie. 01,11,21,…91). Here we have to find trailing knot cords whose cord value ends in 1. (ie. the last eight knot is not added to the penultimate knot).

Code

```{python}
# Collect all trailing 8 knot cords which end in a 1 digit
the_8knotcords_df = figure8knot_fieldmark.relations_df()
pendant_trailing8knot_cords_mask = (the_8knotcords_df['is_subsidiary']==False) & (the_8knotcords_df['eight_knot_type']=='trailing_8knot')
pendant_trailing8knot_cords_df = the_8knotcords_df[pendant_trailing8knot_cords_mask]   
print(f"Number of Pendant Trailing 8 Knot Cords ending in '1': {len(pendant_trailing8knot_cords_df)}")

```

Number of Pendant Trailing 8 Knot Cords ending in '1': 2920

Code

```{python}
ends_in_one_mask = [str(cord_value).endswith('1') for cord_value in pendant_trailing8knot_cords_df['cord_value'].tolist()]
ends_in_one_df = pendant_trailing8knot_cords_df[ends_in_one_mask]
ends_in_one_df.head()
print(f"Number of Trailing 8 Knot Cords ending in '1': {len(ends_in_one_df)}")
```

Number of Trailing 8 Knot Cords ending in '1': 2028

Code

```{python}
kfg_names = ukhipu.lexigraphic_order(list(set(ends_in_one_df['kfg_name'].to_list())))
trailing8knot_1s_cords = []
for kfg_name in kfg_names:
    num_trailing8knot_1s_cords = len(ends_in_one_df[ends_in_one_df['kfg_name']==kfg_name])
    num_pendant_cords = khipu_dict[kfg_name].num_pendant_cords()
    pct_trailing8knot_cords = round(100.0*num_trailing8knot_1s_cords/num_pendant_cords)
    trailing8knot_cord_dict = {'kfg_name':kfg_name,
                               'num_trailing8knot_1s_cords':num_trailing8knot_1s_cords,
                               'num_pendant_cords':num_pendant_cords,
                               'pct_trailing8knot_cords':pct_trailing8knot_cords,
                               }
    trailing8knot_1s_cords.append(trailing8knot_cord_dict)  
trailing8knot_1s_cords_df = pd.DataFrame(trailing8knot_1s_cords)
```

Code

```{python}
print("Statistics for % Trailing 8 Knot Cords ending in 1 vs Pendant Cords")
print(umath.get_statistics(trailing8knot_1s_cords_df['pct_trailing8knot_cords'].to_list()))

fig = (px.scatter(trailing8knot_1s_cords_df, 
                    x='num_pendant_cords',
                    y='num_trailing8knot_1s_cords', 
                    color='pct_trailing8knot_cords', 
                    color_continuous_scale='Bluered_r', 
                    log_x=True, log_y=True,
                    title="By Khipu - # Pendant Cords vs # Sole 8 Knot Cords - Hover over Dot for Detail",
                    labels={'num_pendant_cords':'# Pendant Cords (Log scale)', 
                            'num_trailing8knot_1s_cords':'# Trailing Figure-8-Knot Cords (Log scale)',
                            'pct_trailing8knot_cords': '% Trailing 8 Knot Cords vs Pendant Cords',
                            'kfg_name':'Khipu Name',
                            },
                    width=944, height=944,
                    trendline="ols",
                    hover_data=['num_pendant_cords', 'num_trailing8knot_1s_cords', 'pct_trailing8knot_cords', 'kfg_name'])).show()

```

Statistics for % Trailing 8 Knot Cords ending in 1 vs Pendant Cords
{'min': 0, 'mean': 7.828488372093023, 'median': 6.0, 'max': 58, 'stddev': 6.965320344696113, 'range': 58, 'spread': 0.24018346016193493}

Here we see a closer match - a mean of 7.8% of the pendant cords are trailing 8-knot cords ending in ‘1’, with a standard deviation of 7%, with a skeptically extracted trendline of only 4%. While this might argue against trailing 8 knots being a significant marker, there is this curious issue as to how the existing trailing 1’s gravitate to the edges of a summand range.