// This source code is subject to the terms of the Mozilla Public License 2.0 at https://mozilla.org/MPL/2.0/
// © Bill_Howell
// "$d_web"'CompLangs/PineScript/Howell - SP500 trend 1926-2022.txt'
// view this file in a text editor, with [constant width font, tab = 3 spaces], no line-wrap


//@version=5
indicator("SP500 semi-log trend 1926-2022", overlay=true)


//	**************************
//	relStdDevTrn - relative standard deviation in transformed coorodinates 

//	"$d_web"'economics, markets/SP500/PuetzUWS intnl stocks/semi-log SP500 1926-2022 TV,yahoo,article regression fit.txt' 
//	09Aug2022 last run
//	
//	paramSLRgressL :  
//	+--------------------------------+-------+---------+-------+---------+---------+--------+
//	|symName                         |xTrnPhr|xUnTrnPhr|yTrnPhr|yUnTrnPhr|slope    |constant|
//	+--------------------------------+-------+---------+-------+---------+---------+--------+
//	|SP500 1926-2022 TV,yahoo,article|pass   |?noexpr  |log    |?noexpr  |0.0298315|-56.7736|
//	+--------------------------------+-------+---------+-------+---------+---------+--------+
//	
//	key statistics (in transformed co-ordinates [xTrnPhr, yTrnPhr]) : 
//	+------------+------------+
//	|relStdDevObs|relStdDevTrn|
//	+------------+------------+
//	|    0.333903|   0.0937853|
//	+------------+------------+
//	

relStdDev_plus1 = 1 + 0.0937853


//	**************************
// time conversions  
//	from "$d_web"'CompLangs/PineScript/PuetzUWS IntlStkIdxs multi-fractals.txt'

var float t_year_to_millisecs  = 365.25*24*60*60*1000

yrFrac = time / t_year_to_millisecs + 1970


//	**************************
//	trendlines

SP500_base_trender = math.pow(10, -56.7736 + (0.029831 * yrFrac)) 
SP500_plus_1stdDev = SP500_base_trender * relStdDev_plus1
SP500_less_1stdDev = SP500_base_trender / relStdDev_plus1


//	plots
//	Note that this plot may "scrunch" the actual SP500 plot if is on the same axis
//		if not on the same axis, there will be an annoying offset

plot(SP500_base_trender, color=color.black, linewidth=2, title="SP500_base_trender")
plot(SP500_plus_1stdDev, color=color.gray , linewidth=1, title="SP500_plus_1stdDev")
plot(SP500_less_1stdDev, color=color.gray , linewidth=1, title="SP500_less_1stdDev")




// enddoc