prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What type of distribution do the tail distributions of price returns follow?

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Context 1: Paper Info

Title: Age and market capitalization drive large price variations of cryptocurrencies
Publish Date: 23 Feb 2023
Author List: 

Figure

Figure 3  Illustration of different effects of age and market capitalization on power-law exponents of cryptocurrencies (a) Posterior probability distributions of the linear coefficients associated with the effects of age [p(A)] and (b) the effects of market capitalization [p(C)] on power-law exponents related to large positive returns Panels (c) and (d) show the analogous distributions for the association with power-law exponents related to large negative returns In all panels, the different curves show the distributions for each of the top 20 cryptoassets by market capitalization Cryptocurrencies significantly affected by age or market capitalization are highlighted in boldface, and the numbers between brackets show their positions in the market capitalization rank 
Figure S5 There is more probability mass in the positive tail than in the negative tail of price returns (a) Probability distributions of the lower cut-offs (r min ) obtained by applying the Clauset-Shalizi-Newman method to positive (blue) and negative (red) returns The vertical dashed lines indicate the median values of r min for positive and negative returns (b) Probability distributions of 90th percentiles (r 90 ) estimated from the power-law models adjusted to positive (blue) and negative (red) returns The vertical dashed lines indicate the median values of r 90 for positive and negative returns (c) Probability distributions of the fraction of weeks that r 90 estimated from positive returns (r + 90 ) is larger than r 90 estimated from negative returns (r − 90 ) This fraction is calculated only for weeks in which the power-law hypothesis is not rejected for both tails The percentage of cryptoassets for which r + 90 > r − 90 is shown in the panels The first column of panels depicts the results when considering data from all cryptocurrencies, while the second and third columns present the results for the top 2000 and top 200 cryptocurrencies by market capitalization, respectively 
Figure S7 Robustness of the results of Fig  2(b)-(d) against considering only cryptocurrencies with fraction of rejection f r < 0 1 Panels (a) and (b) show the same distributions of Fig  S4 but after filtering out all time series of cryptocurrencies with fraction of rejections f r ≥ 0 1 As in the case related to sampling issues, we observe that these distributions barely change when considering only cryptocurrencies with f r < 0 1 Indeed, the distributions in this figure are not significantly distinguishable from their counterparts in Fig  S4 (two-sample Kolmogorov-Smirnov test, p > 0 05) 

abstract

Cryptocurrencies are considered the latest innovation in finance with considerable impact across social, technological, and economic dimensions  This new class of financial assets has also motivated a myriad of scientific investigations focused on understanding their statistical properties, such as the distribution of price returns 
However, research so far has only considered Bitcoin or at most a few cryptocurrencies, whilst ignoring that price returns might depend on cryptocurrency age or be influenced by market capitalization  Here, we therefore present a comprehensive investigation of large price variations for more than seven thousand digital currencies and explore whether price returns change with the coming-of-age and growth of the cryptocurrency market 
We find that tail distributions of price returns follow power-law functions over the entire history of the considered cryptocurrency portfolio, with typical exponents implying the absence of characteristic scales for price variations in about half of them  Moreover, these tail distributions are asymmetric as positive returns more often display smaller exponents, indicating that large positive price variations are more likely than negative ones 
Our results further reveal that changes in the tail exponents are very often simultaneously related to cryptocurrency age and market capitalization or only to age, with only a minority of cryptoassets being affected just by market capitalization or neither of the two quantities  Lastly, we find that the trends in power-law exponents usually point to mixed directions, and that large price variations are likely to become less frequent only in about 28% of the cryptocurrencies as they age and grow in market capitalization 
Since the creation of Bitcoin in 2008 , various different cryptoassets have been developed and are now considered to be at the cutting edge of innovation in finance   These digital financial assets are vastly diverse in design characteristics and intended purposes, ranging from peer-to-peer networks with underlying cash-like digital currencies (e g 
Bitcoin) to general-purpose blockchains transacting in commodity-like digital assets (e g  Ethereum), and even to cryptoassets that intend to replicate the price of conventional assets such as the US dollar or gold (e g  Tether and Tether Gold)   With more than nine thousand cryptoassets as of 2022 , the total market value of cryptocurrencies has grown massively to a staggering $2 trillion peak in 2021  
Despite long-standing debates over the intrinsic value and legality of cryptoassets , or perhaps even precisely due to such controversies, it is undeniable that cryptocurrencies are increasingly attracting the attention of academics, investors, and central banks, around the world   Moreover, these digital assets have been at the forefront of sizable financial gains and losses in recent years , they have been recognized as the main drivers of the brand-new phenomena of cryptoart and NFTs , but also as facilitators of illegal activities, such as money laundering and dark trade  
Financial research dedicated Our results are based on daily price time series of 7111 cryptocurrencies that comprise a significant part of all currently available cryptoassets (see Methods for details)  From these price series, we have estimated their logarithmic returns 2/16 Log-return, r )  The black horizontal arrow represents a given position of the expanding time window (at t = 2004 days) used to sample the return series over the entire history of Bitcoin 
This time window expands in weekly steps (seven time series observations), and for each position, we separate the positive (blue) from the negative (red) price returns  The gray line illustrates observations that will be included in future positions of the expanding time window (t > 2004)  (b) Survival functions or the complementary cumulative distributions of positive (blue) and negative (red) price returns within the expanding time window for t = 2004 days and above the lower bound of the power-law regime estimated from the Clauset-Shalizi-Newman method  
The dashed lines show the adjusted power-law functions, p(r) ∼ r −α , with α = 4 5 for positive returns and α = 3 0 for negative returns  (c) Time series of the power-law exponents α t for the positive (blue) and negative (red) return distributions obtained by expanding the time window from the hundredth observation (t = 100) to the latest available price return of Bitcoin 
The circular markers represent the values for the window position at t = 2004 days and the dashed lines indicate the median of the power-law exponents ( α+ = 4 50 for positive returns and α− = 2 99 for negative returns)  (d) Time series of the p-values related to the power-law hypothesis of positive (blue) and negative (red) price returns for every position of the expanding time window 
The dashed line indicates the threshold (p = 0 1) above which the power-law hypothesis cannot be rejected  For Bitcoin, the power-law hypothesis is never rejected for positive returns (fraction of rejection f r = 0) and rejected in only 4% of the expanding time window positions (fraction of rejection f r = 0 04) 
where x t represents the price of a given cryptocurrency at day t  All return time series in our analysis have at least 200 observations (see Supplementary Figure for the length distribution)  Figure (a) illustrates Bitcoin's series of daily returns  To investigate whether and how returns have changed over the aging and growing processes of all cryptocurrencies, we sample all time series of log-returns using a time window that expands in weekly steps (seven time series observations), starting from the hundredth observation to the latest return observation 
In each step, we separate the positive from the negative return values and estimate their power-law behavior using the Clauset-Shalizi-Newman method   Figure (a) further illustrates this procedure, where the vertical dashed line represents a given position of the time window (t = 2004 days), the blue and red lines indicate positive and negative returns, respectively, and the gray lines show the return observations that will be included in the expanding time window in future steps 
Moreover, Fig  (b) shows the corresponding survival functions (or complementary cumulative distributions) for the positive (blue) and negative (red) returns of Bitcoin within the time window highlighted in Fig  (a)  These survival functions correspond to return values above the lower bound of the power-law regime (r min ) and dashed lines in Fig  (b) show the power-law functions adjusted to data, that is,
with α = 4 5 for the positive returns and α = 3 0 for the negative returns in this particular position of the time window (t = 2004 days)  We have further verified the goodness of the power-law fits using the approach proposed by Clauset et al  (see also Preis et al  )  As detailed in the Methods section, this approach consists in generating several synthetic samples under the power-law hypothesis, adjusting these simulated samples, and estimating the fraction of times the Kolmogorov-Smirnov distance between the adjusted power-law and the synthetic samples is larger than the value calculated from the empirical data 
This fraction defines a p-value and allows us to reject or not the power-law hypothesis of the return distributions under a given confidence level  Following Refs  we consider the more conservative 90% confidence level (instead of the more lenient and commonly used 95% confidence level), rejecting the power-law hypothesis when p-value ≤ 0 1 
For the particular examples in Fig  (b), the p-values are respectively 1 00 and 0 17 for the positive and negative returns, and thus we cannot reject the power-law hypotheses  After sampling the entire price return series, we obtain time series for the power-law exponents (α t ) associated with positive and negative returns as well as the corresponding p-values time series for each step t of the expanding time window 
These time series allow us to reconstruct the aging process of the return distributions over the entire history of each cryptoasset and probe possible time-dependent patterns  Figures ) and 1(d) show the power-law exponents and p-values time series for the case of Bitcoin  The power-law hypothesis is never rejected for positive returns and rarely rejected for negative returns (about 4% of times) 
Moreover, the power-law exponents exhibit large fluctuations at the beginning of the time series and become more stable as Bitcoin matures as a financial asset (a similar tendency as reported by Begušić et al  )  The time evolution of these exponents further shows that the asymmetry between positive and negative returns observed in Fig  ) is not an incidental feature of a particular moment in Bitcoin's history 
Indeed, the power-law exponent for positive returns is almost always larger than the exponent for negative returns, implying that large negative price returns have been more likely to occur than their positive counterparts over nearly the entire history of Bitcoin covered by our data  However, while the difference between positive and negative exponents has approached a constant value, both exponents exhibit an increasing trend, indicating that large price variations are becoming less frequent with the coming-of-age of Bitcoin 
The previous analysis motivates us to ask whether the entire cryptocurrency market behaves similarly to Bitcoin and what other common patterns digital currencies tend to follow  To start answering this question, we have considered the p-values series of all cryptocurrencies to verify if the power-law hypothesis holds in general 
Figure (a) shows the percentage of cryptoassets rejecting the power-law hypothesis in at most a given fraction of the weekly positions of the expanding time window ( f r )  Remarkably, the hypothesis that large price movements (positive or negative) follow a power-law distribution is never rejected over the entire history of about 70% of all digital currencies in our dataset 
This analysis also shows that only ≈2% of cryptocurrencies reject the power-law hypothesis in more than half of the positions of the expanding time window ( f r ≥ 0 5)  For instance, considering a 10% threshold as a criterion ( f r ≤ 0 1), we find that about 85% of cryptocurrencies have return distributions adequately modeled by power laws 
Increasing this threshold to a more lenient 20% threshold ( f r ≤ 0 2), we find large price movements to be power-law distributed for about 91% of cryptocurrencies  These results thus provide strong evidence that cryptoassets, fairly generally, present large price movements quite well described by power-law distributions 
Moreover, this conclusion is robust when starting the expanding window with a greater   Large price movements are power-law distributed over the entire history of most cryptocurrencies with median values typically smaller than those found for traditional assets  (a) Percentage of cryptoassets rejecting the power-law hypothesis for large positive (blue) or negative (red) price returns in at most a given fraction of the weekly positions of the expanding time window ( f r ) used to sample the return series 
Remarkably, 68% of all 7111 digital currencies are compatible with the power-law hypothesis over their entire history, and about 91% of them reject the power-law hypothesis in less than 20% of the positions of the expanding time window ( f r ≤ 0 2)  (b) Probability distributions obtained via kernel density estimation of the median values of the power-law exponents along the history of each digital currency 
The blue curve shows the distribution of the median exponents related to positive returns ( α+ ) and the red curve does the same for negative returns ( α− )  The medians of α+ and α− are indicated by vertical dashed lines  Panels (c) and (d) show the distributions of these median exponents when considering the top 2000 and the top 200 cryptocurrencies by market capitalization, respectively 
We observe that the distributions of α+ and α− tend to shift toward larger values when considering the largest cryptoassets  number of return observations (between 100 and 300 days) and filtering out cryptoassets with missing observations (Supplementary Figures )  Still, it is worth noticing the existence of a few cryptoassets (9 of them) with relatively small market capitalization (ranking below the top 1000) for which the power-law hypothesis is always rejected (Supplementary Table ) 
Having verified that large price movements in the cryptocurrency market are generally well-described by powerlaw distributions, we now focus on the power-law exponents that typically characterize each cryptoasset  To do so, we select all exponent estimates over the entire history of each digital asset for which the power-law hypothesis is not rejected and calculate their median values for both the positive ( α+ ) and negative ( α− ) returns 
The dashed lines in Fig  ) show these median values for Bitcoin where α+ = 4 50 and α− = 2 99  It is worth noticing that the variance of large price movements σ 2 is finite only for α > 3, as the integral σ 2 ∼ ∞ r min r 2 p(r)dr diverges outside this interval  Thus, while the typical variance of large positive returns is finite for Bitcoin, negative returns are at the limit of not having a typical scale and are thus susceptible to much larger variations 
Figure shows the probability distribution for the median power-law exponents of all cryptoassets grouped by large positive and negative returns  We note that the distribution of typical power-law exponents associated with large positive returns is shifted to smaller values when compared with the distribution of exponents related to large negative returns 
The medians of these typical exponents are respectively 2 78 and 3 11 for positive and negative returns  This result suggests that the asymmetry in large price movements we have observed for Bitcoin is an overall feature of the cryptocurrency market  By calculating the difference between the typical exponents related to positive and negative large returns (∆α = α+ − α− ) for each digital currency, we find that about 2/3 of cryptocurrencies have α+ < α− (see Supplementary Figure for the probability distribution of ∆α) 
Thus, unlike Bitcoin, most cryptocurrencies have been more susceptible to large positive price variations than negative ones  While this asymmetry in the return distributions indicates that extremely large price variations tend to be positive, it does not necessarily imply positive price variations are more common for any threshold in the return values 
This happens because the fraction of events in each tail is also related to the lower bound of the power-law regime (r min )  However, we have found the distribution of r min to be similar among the positive and negative returns [Supplementary Figure ]  The distribution of high percentile scores (such as the 90th percentile) is also shifted to larger values for positive returns [Supplementary Figure ] 
Moreover, this asymmetry in high percentile scores related to positive and negative returns is systematic along the evolution of the power-law exponents [Supplementary Figure ]  These results thus indicate that there is indeed more probability mass in the positive tails than in the negative ones, a feature that likely reflects the current expansion of the cryptocurrency market as a whole 
The distributions in Fig  ) also show that large price variations do not have a finite variance for a significant part of cryptoassets, that is, α+ ≤ 3 for 62% of cryptocurrencies and α− ≤ 3 for 44% of cryptocurrencies  A significant part of the cryptocurrency market is thus prone to price variations with no typical scale 
Intriguingly, we further note the existence of a minority group of cryptoassets with α+ ≤ 2 (7%) or α− ≤ 2 (3%)  These cryptocurrencies, whose representative members are Counos X (CCXX, rank 216) with α − = 1 96 and α + = 1 84 and Chainbing (CBG, rank 236) with α + = 1 87, are even more susceptible to extreme price variations as one cannot even define the average value µ for large price returns, as the integral µ ∼ ∞ r min rp(r)dr diverges for α ≤ 2  We have also replicated the previous analysis when considering cryptocurrencies in the top 2000 and top 200 rankings of market capitalization (as of July 2022) 
Figures ) and 2(d) show the probability distribution for the median power-law exponents of these two groups  We observe that these distributions are more localized (particularly for the top 200) than the equivalent distributions for all cryptocurrencies  The fraction of cryptocurrencies with no typical scale for large price returns ( α+ ≤ 3 and α− ≤ 3) is significantly lower in these two groups compared to all cryptocurrencies 
In the top 2000 cryptocurrencies, 51% have α+ ≤ 3 and 26% have α− ≤ 3  These fractions are even smaller among the top 200 cryptocurrencies, with only 44% and 15% not presenting a typical scale for large positive and negative price returns, respectively  We further observe a decrease in the fraction of cryptoassets for which the average value for large price returns is not even finite, as only 2% and 1% of top 2000 cryptoassets have α+ ≤ 2 and α− ≤ 2  This reduction is more impressive among the top 200 cryptocurrencies as only the cryptoasset Fei USD (FEI, rank 78) has α+ = 1 97 and none is characterized by α− ≤ 2  The medians of α+ and α− also increase from 2 78 and 3 11 for all cryptocurrencies to 2 98 and 3 35 for the top 2000 and to 3 08 and 3 58 for the top 200 cryptocurrencies 
Conversely, the asymmetry between positive and negative large price returns does not differ much among the three groups, with the condition α+ < α− holding only for a slightly larger fraction of top 2000 (69 1%) and top 200 (70 6%) cryptoassets compared to all cryptocurrencies (66 4%)  Moreover, all these patterns are robust when filtering out time series with sampling issues or when considering only cryptoassets that stay compatible with the power-law hypothesis in more than 90% of the positions of the expanding time window (Supplementary Figures ) 
We also investigate whether the patterns related to the median of the power-law exponents differ among groups of cryptocurrencies with different designs and purposes  To do so, we group digital assets using the 50 most common tags in our dataset (e g  "bnb-chain", "defi", and "collectibles-nfts") and estimate the probability distributions of the median exponents α+ and α− (Supplementary Figures ) 
These results show that design and purpose affect the dynamics of large price variations in the cryptocurrency market as the medians of typical exponents range from 2 4 to 3 7 among the groups  The lowest values occur for cryptocurrencies tagged as "doggone-doggerel" (medians of α+ and α− are 2 38 and 2 83), "memes" (2 41 and 2 87), and "stablecoin" (2 65 and 2 79) 
Digital currencies belonging to the first two tags overlap a lot and have Dogecoin (DOGE, rank 9) and Shiba Inu (SHIB, rank 13) as the most important representatives  Cryptoassets with these tags usually have humorous characteristics (such as an Internet meme) and several have been considered as a form of pump-and-dump scheme , a type of financial fraud in which false statements artificially inflate asset prices so the scheme operators sell their overvalued cryptoassets 
Conversely, cryptoassets tagged as "stablecoin" represent a class of cryptocurrencies designed to have a fixed exchange rate to a reference asset (such as a national currency or precious metal)   While the price of stablecoins tends to stay around the target values, their price series are also marked by sharp variations, which in turn are responsible for their typically small power-law exponents 
This type of cryptoasset has been shown to be prone to failures , such as the recent examples of TerraUSD (UST) and Tron's USDD (USDD) that lost their pegs to the US Dollar producing large variations in their price series  The asymmetry between positive and negative large returns also emerges when grouping the cryptocurrencies using their tags 
All 50 tags have distributions of α+ shifted to smaller values when compared with the distributions of α− , with differences between their medians ranging from −0 74 ("okex-blockdream-ventures-portfolio") to −0 14 ("stablecoin")  Indeed, only four ('stablecoin", "scrypt", "fantom-ecosystem" and "alameda-research-portfolio") out of the fifty groupings have both distributions indistinguishable under a two-sample Kolmogorov-Smirnov test (p-value > 0 05) 
Focusing now on the evolution of the power-law exponents quantified by the time series α t for positive and negative returns, we ask whether these exponents present particular time trends  For Bitcoin [Fig  )], α t seems to increase with time for both positive and negative returns  At the same time, the results of Fig  also suggest that market capitalization affects these power-law exponents 
To verify these possibilities, we assume the power-law exponents (α t ) to be linearly associated with the cryptocurrency's age (y t , measured in years) and the logarithm of market capitalization (log c t )  As detailed in the Methods section, we frame this problem using a hierarchical Bayesian model 
This approach assumes that the linear coefficients associated with the effects of age (A) and market capitalization (C) of each digital currency are drawn from distributions with means µ A and µ C and standard deviations σ A and σ C , which are in turn distributed according to global distributions representing the overall impact of these quantities on the cryptocurrency market 
The Bayesian inference process consists of estimating the posterior probability distributions of the linear coefficients for each cryptocurrency as well as the posterior distributions of µ A , µ C , σ A , and σ C , allowing us to simultaneously probe asset-specific tendencies and overall market characteristics 
Moreover, we restrict this analysis to the 2140 digital currencies having more than 50 observations of market capitalization concomitantly to the time series of the power-law exponents in order to have enough data points for detecting possible trends  When considering the overall market characteristics, we find that the 94% highest density intervals for µ A ([-0 01, 0 06] for positive and [-0 02, 0 03] for negative returns) and µ C ([-0 02, 0 03] for positive and [-0 001, 0 04] for negative returns) include the zero (see Supplementary Figure for their distributions) 
Thus, there is no evidence of a unique overall pattern for the association between the power-law exponents and age or market capitalization followed by a significant part of the cryptocurrency market  Indeed, the 94% highest density intervals for σ A ([0 87, 0 93] for positive and [0 63, 0 70] for negative returns) and σ C ([0 57, 0 61] for positive and [0 49, 0 52] for negative returns) indicate that the cryptocurrency market is highly heterogeneous regarding the evolution of power-law exponents associated with large price variations (see Supplementary Figure for the distributions of σ A and σ C )  Figure illustrates these heterogeneous behaviors by plotting the posterior probability distributions for the linear coefficients associated with the effects of age (A) and market capitalization (C) for the top 20 digital assets, where cryptocurrencies which are significantly affected (that is, the 94% highest density intervals for A or C do not include the zero) by these quantities are highlighted in boldface 
Even this small selection of digital  currencies already presents a myriad of patterns  First, we observe that the power-law exponents of a few top 20 cryptocurrencies are neither correlated with age nor market capitalization  That is the case of Shiba Inu (SHIB, rank 13) and Dai (DAI, rank 11) for both positive and negative returns, UNUS SED LEO (LEO, rank 18) and Polkadot (DOT, rank 12) for the positive returns, and USDCoin (USDC, rank 4) and Solana (SOL, rank 9) for negative returns 
There are also cryptocurrencies with exponents positively or negatively correlated only with market capitalization  Examples include Tether (USDT, rank 3) and Dogecoin (DOGE, rank 10), for which the power-law exponents associated with positive returns increase with market capitalization, and Binance USD (BUSD, rank 6), for which power-law exponents associated with positive and negative returns decrease with market capitalization 
We also observe cryptocurrencies for which age and market capitalization simultaneously affect the power-law exponents  Polygon (MATIC, rank 14) is an example where the power-law exponents associated with positive returns tend to increase with age and decrease with market capitalization  Finally, there are also cryptocurrencies with power-law exponents only associated with age 
That is the case of Bitcoin (BTC, rank 1), Ethereum (ETH, rank 2), and Cardano (ADA, rank 8), for which the power-law exponents related to positive and negative returns increase with age, but also the case of Uniswap (UNI, rank 19), for which the exponents decrease with age  Figure systematically extends the observations made for the top 20 cryptoassets to all 2140 digital currencies for which we have modeled the changes in the power-law exponents as a function of age and market capitalization 
First, we note that only 10% of cryptocurrencies have power-law exponents not significantly affected by age and market capitalization  The vast majority (90%) displays some relationship with these quantities  However, these associations are as varied as the ones we have observed for the top 20 cryptoassets 
About 52% of cryptocurrencies have power-law exponents simultaneously affected by age and market capitalization  In this group, these quantities simultaneously impact the exponents related to positive and negative returns of 34% of cryptoassets, whereas the remainder is affected only in the positive tail (9%) or only in the negative tail (9%) 
Moving back in the hierarchy, we find that the power-law exponents of 32% of cryptocurrencies are affected only by age while a much minor fraction (6%) is affected only by market capitalization  Within the group only affected by age, we observe that the effects are slightly more frequent only on the exponents related to negative returns (12%), compared to cases where effects are restricted only to positive returns (10%) or simultaneously affect both tails (10%) 
Finally, within the minor group only affected by market capitalization, we note that associations more frequently involve only exponents related to negative returns (3%) compared to the other two cases (2% only positive returns and 1% for both positive and negative returns)  Beyond the previous discussion about whether positive or negative returns are simultaneously or individually affected by age and market capitalization, we have also categorized the direction of the trend imposed by these two quantities on the power-law exponents 
Blue rectangles in Fig  represent the fraction of relationships for which increasing age or market capitalization (or both) is associated with a raise in the power-law exponents  About 28% of all cryptocurrencies exhibit this pattern in which large price variations are expected to occur less frequently as they grow and age 
Conversely, the red rectangles in Fig  depict the fraction of relationships for which increasing age or market capitalization (or both) is associated with a reduction in the power-law exponents  This case comprises about 25% of all cryptocurrencies for which large price variations are likely to become more frequent as they grow in market capitalization and age 
Still, the majority of associations represented by green rectangles refer to the case where the effects of age and market capitalization point in different directions (e g  exponents increasing with age while decreasing with market capitalization)  About 36% of cryptocurrencies fit this condition which in turn contributes to consolidating the cumbersome hierarchical structure of patterns displayed by cryptocurrencies regarding the dynamics of large price variations 
This complex picture is not much different when considering only cryptocurrencies in the top 200 market capitalization rank (Supplementary Figure )  However, we do observe an increased prevalence of patterns characterized by exponents that rise with age and market capitalization (37%), suggesting that large price variations are becoming less frequent among the top 200 cryptocurrencies than in the overall market 
)  Each of the previous three levels is further classified regarding whether both positive and negative returns are simultaneously affected or whether the effect involves only positive or only negative returns  Finally, the former levels are classified regarding whether the power-law exponents increase, decrease or have a mixed trend with the predictive variables 
Overall, 36% of the associations are classified as mixed trends (green rectangles), 28% are increasing trends (blue rectangles), and 26% are decreasing trends (red rectangles)  We have studied the distributions of large price variations of a significant part of the digital assets that currently comprise the entirety of the cryptocurrency market 
Unlike previous work, we have estimated these distributions for entire historical price records of each digital currency, and we have identified the patterns under which the return distributions change as cryptoassets age and grow in market capitalization  Similarly to conventional financial assets , our findings show that the return distributions of the vast majority of cryptoassets have tails that are described well by power-law functions along their entire history 
The typical power-law exponents of cryptocurrencies (α ∼ 3) are, however, significantly smaller than those reported for conventional assets (α ∼ 4)   This feature corroborates the widespread belief that cryptoassets are indeed considerably more risky for investments than stocks or other more traditional financial assets 
Indeed, we have found that about half of the cryptocurrencies in our analysis do not have a characteristic scale for price variations, and are thus prone to much higher price variations than those typically observed in stock markets  On the upside, we have also identified an asymmetry in the power-law exponents for positive and negative returns in about 2/3 of all considered cryptocurrencies, such that these exponents are smaller for positive than they are for negative returns 
This means that sizable positive price variations have generally been more likely to occur than equally sizable negative price variations, which in turn may also reflect the recent overall expansion of the cryptocurrency market  Using a hierarchical Bayesian linear model, we have also simultaneously investigated the overall market characteristics and asset-specific tendencies regarding the effects of age and market capitalization on the power-law exponents 
We have found that the cryptocurrency market is highly heterogeneous regarding the trends exhibited by each cryptocurrency; however, only a small fraction of cryptocurrencies (10%) have power-law exponents neither correlated with age nor market capitalization  These associations have been mostly ignored by the current literature and are probably related to the still-early developmental stage of the cryptocurrency market as a whole 
Overall, 36% of cryptocurrencies present trends that do not systematically contribute to increasing or decreasing their power-law exponents as they age and grow in market capitalization  On the other hand, for 26% of cryptocurrencies, aging and growing market capitalization are both associated with a reduction in their power-law exponents, thus contributing to the rise in the frequency of large price variations in their dynamics 
Only about 28% of cryptocurrencies present trends in which the power-law exponents increase with age and market capitalization, favoring thus large price variations to become less likely  These results somehow juxtapose with findings about the increasing informational efficiency of the cryptocurrency market  
In fact, if on the one hand the cryptocurrency market is becoming more informationally efficient, then on the other our findings indicate that there is no clear trend toward decreasing the risks of sizable variations in the prices of most considered cryptoassets  In other words, risk and efficiency thus appear to be moving towards different directions in the cryptocurrency market 
To conclude, we hope that our findings will contribute significantly to the better understanding of the dynamics of large price variations in the cryptocurrency market as a whole, and not just for a small subset of selected digital assets, which is especially relevant due to the diminishing concentration of market capitalization among the top digital currencies, and also because of the considerable impact these new assets may have in our increasingly digital economy 
Our results are based on time series of the daily closing prices (in USD) for all cryptoassets listed on CoinMar-ketCap (coinmarketcap com) as of 25 July 2022 [see Supplementary Figure (a) for a visualization of the increasing number cryptoassets listed on CoinMarketCap since 2013]  These time series were automatically gathered using the cryptoCMD Python package and other information such as the tags associated with each cryptoasset were obtained via the CoinMarketCap API  
In addition, we have also obtained the daily market capitalization time series (in USD) from all cryptoassets which had this information available at the time  Earliest records available from CoinMarketCap date from 29 April 2013 and the latest records used in our analysis correspond to 25 July 2022  Out of 9943 cryptocurrencies, we have restricted our analysis to the 7111 with at least 200 price-return observations 
The median length of these time series is 446 observations [see the distribution of series length in Supplementary Figure   We have estimated the power-law behavior of the return distributions by applying the Clauset-Shalizi-Newman method to the return time series r t   In particular, we have sampled each of these time series using an expanding time window that starts at the hundredth observation and grows in weekly steps (seven data points each step) 
For each position of the expanding time window, we have separated the positive returns from the negative ones and applied the Clauset-Shalizi-Newman method to each set  This approach consists of obtaining the maximum likelihood estimate for the power-law exponent, α = 1 + n/ (∑ n t=1 ln r t /r min ) , where r min is the lower bound of the power-law regime and n is the number of (positive or negative) return observations in the power-law regime for a given position of the expanding time window 
The value r min is estimated from data by minimizing the Kolmogorov-Smirnov statistic between the empirical distribution and the power-law model  The Clauset-Shalizi-Newman method yields an unbiased and consistent estimator , in a sense that as the sample increases indefinitely, the estimated power-law exponent converges in distribution to the actual value 
Moreover, we have used the implementation available on the powerlaw Python package   In addition to obtaining the power-law exponents, we have also verified the adequacy of the power-law hypothesis using the procedure originally proposed by Clauset et al  as adapted by Preis et al    This procedure consists of generating synthetic samples under the power-law hypothesis with the same properties of the empirical data under analysis (that is, same length and parameters α and r min ), adjusting the simulated data with the power-law model via the Clauset-Shalizi-Newman method, and calculating the Kolmogorov-Smirnov statistic (κ syn ) between the distributions obtained from the simulated samples and the adjusted power-law model 
Next, the values of κ syn are compared to the Kolmogorov-Smirnov statistic calculated between empirical data and the power-law model (κ)  Finally, a p-value is defined by calculating the fraction of times for which κ syn > κ  We have used one thousand synthetic samples for each position of the expanding time window and the more conservative 90% confidence level (instead of the more lenient and commonly used 95% confidence level), such that the power-law hypothesis is rejected whenever p-value ≤ 0 1 
We have estimated the effects of age and market capitalization on the power-law exponents associated with positive or negative returns of a given cryptocurrency using the linear model where α t represents the power-law exponent, log c t is the logarithm of the market capitalization, and y t is the age (in years) of the cryptocurrency at t-th observation 
Moreover, K is the intercept of the association, while C and A are linear coefficients quantifying the effects of market capitalization and age, respectively  Finally, N (µ, σ ) stands for the normal distribution with mean µ and standard deviation σ , such that the parameter ε accounts for the unobserved determinants in the dynamics of the power-law exponents 
We have framed this problem using the hierarchical Bayesian approach such that each power-law exponent α t is nested within a cryptocurrency with model parameters considered as random variables normally distributed with parameters that are also random variables  Mathematically, for each cryptocurrency, we have
12/16 where µ K , σ K , µ C , σ C , µ A , and σ A are hyperparameters  These hyperparameters are assumed to be distributed according to distributions that quantify the overall impact of age and market capitalization on the cryptocurrency market as a whole  We have performed this Bayesian regression for exponents related to positive and negative returns separately, and used noninformative prior and hyperprior distributions in order not to bias the posterior estimation  
Specifically, we have considered and ε ∼ U (0, 10 2 ) , where U (a, b) stands for the uniform distribution in the interval [a, b] and Inv−Γ(θ , γ) represents the inverse gamma distribution with shape and scale parameters θ and γ, respectively  For the numerical implementation, we have relied on the PyMC Python package and sampled the posterior distributions via the gradient-based Hamiltonian Monte Carlo no-U-Turn-sampler method 
We have run four parallel chains with 2500 iterations each (1000 burn-in samples) to allow good mixing and estimated the Gelman-Rubin convergence statistic (R-hat) to ensure the convergence of the sampling approach (R-hat was always close to one)  In addition, we have also verified that models describing the power-law exponents as a function of only age (C → 0 in Eq  3) or only market capitalization (A → 0 in Eq  3) yield significantly worse descriptions of our data as quantified by the Widely Applicable Information Criterion (WAIC) and the Pareto Smoothed Importance Sampling Leave-One-Out cross-validation (PSIS-LOO) (see Supplementary Table )  ) is larger than r 90 estimated from negative returns (r − 90 ) 
This fraction is calculated only for weeks in which the power-law hypothesis is not rejected for both tails  The percentage of cryptoassets for which r + 90 > r − 90 is shown in the panels  The first column of panels depicts the results when considering data from all cryptocurrencies, while the second and third columns present the results for the top 2000 and top 200 cryptocurrencies by market capitalization, respectively 
Sampling issues refer to missing data and problems caused by prices of cryptoassets decreasing to zero  We note that these distributions barely change when considering only cryptocurrencies without any sampling issue  Indeed, the distributions in this figure are not significantly distinguishable from their counterparts in Fig  (two-sample Kolmogorov-Smirnov test, p > 0 05) 
)  Each of the previous three levels is further classified regarding whether both positive and negative returns are simultaneously affected or whether the effect involves only positive or only negative returns  Finally, the former levels are classified regarding whether the power-law exponents increase, decrease or have a mixed trend with the predictive variables 
Overall, 35% of the associations are classified as mixed trends (green rectangles), 37% are increasing trends (blue rectangles), and 18% are decreasing trends (red rectangles) 
Context 3: McPherson County (standard abbreviation: MP) is a county located in the U S  state of Kansas   As of the 2020 census, the county population was 30,223  The largest city and county seat is McPherson  The county is named for Civil War General James B  McPherson 

History

Early history

For many millennia, the Great Plains of North America was inhabited by nomadic Native Americans  From the 16th century to 18th century, the Kingdom of France claimed ownership of large parts of North America  In 1762, after the French and Indian War, France secretly ceded New France to Spain, per the Treaty of Fontainebleau  In 1802, Spain returned most of the land to France, but keeping title to about 7,500 square miles 

In 1803, most of the land for modern day Kansas was acquired by the United States from France as part of the 828,000 square mile Louisiana Purchase for 2 83 cents per acre  In 1848, after the Mexican–American War, the Treaty of Guadalupe Hidalgo with Spain brought into the United States all or part of land for ten future states, including southwest Kansas  In 1854, the Kansas Territory was organized, then in 1861 Kansas became the 34th U S  state 

19th century

From the 1820s to 1870s, the Santa Fe Trail passed through, what is now McPherson County  The trail entered the county, east of Canton, then south of Galva, then north of Inman, and west towards Lyons  In 1855, Charles O  Fuller established a ranch adjacent to the Running Turkey Creek Crossing about two miles south and one mile east of Galva  Fuller's Ranch provided accommodations for travelers on the Santa Fe Trail and was probably the first white settlement in McPherson County 

Peketon County was established in 1860, by the passage of a bill by S  N  Wood:  An act to establish Peketon County  Section 1  - That all that territory west of the sixth principal meridian and south of Township 16, in Kansas Territory, be and the same is hereby erected into a county, to be known by the name of Peketon County  On February 17, 1865, Peketon County was abolished, and McPherson County was made a part of Marion County, which extended from the west line of Chase County to the present western boundary of Kansas 

In 1868, Solomon Stephens and L  N  Holmberg were appointed Justices of the Peace—the first officers in what is now McPherson County  The next year (1869) occurred the first election for the township, now the county of McPherson  McPherson was regularly organized as a county in the spring of 1870, a mass meeting being held at Sweadal  Sweadal, the county seat thus selected, was located about one mile and a half southwest of the present site of Lindsborg  In September, however, the County Commissioners resolved to meet at the latter place, McPherson which had already been located some two years 

In April, 1873, a petition was filed for the county seat re-location  It was signed by 483 voters, and a special election was accordingly ordered for June 10  Upon that day, McPherson received 605 votes, New Gottland 325, King City 3 and Lindsborg 1; McPherson's majority over all, 276  In May the McPherson Town Company had offered, as an inducement for the location of the county seat at this point, the free use of rooms for ten years, and the donation of two squares of land on the town site  The offer was accepted the next month, the County Commissioners selecting blocks 56 and 65  Thus the county seat was established at McPherson and has remained since 

As early as 1875, city leaders of Marion held a meeting to consider a branch railroad from Florence  In 1878, Atchison, Topeka and Santa Fe Railway and parties from Marion County and McPherson County chartered the Marion and McPherson Railway Company   In 1879, a branch line was built from Florence to McPherson, in 1880 it was extended to Lyons, in 1881 it was extended to Ellinwood  The line was leased and operated by the Atchison, Topeka and Santa Fe Railway  The line from Florence to Marion, was abandoned in 1968  In 1992, the line from Marion to McPherson was sold to Central Kansas Railway  In 1993, after heavy flood damage, the line from Marion to McPherson was abandoned



ground_truth: Power-law functions.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What algorithm is engaged in the PLMS-PPIC method?

Read the following text and answer briefly.

Context 2: \section{Introduction}\label{S1}

The multiple access interferences (MAI) is the root of user
limitation in CDMA systems \cite{R1,R3}  The parallel least mean
square-partial parallel interference cancelation (PLMS-PPIC) method
is a multiuser detector for code division multiple access (CDMA)
receivers which reduces the effect of MAI in bit detection  In this
method and similar to its former versions like LMS-PPIC \cite{R5}
(see also \cite{RR5}), a weighted value of the MAI of other users is
subtracted before making the decision for a specific user in
different stages \cite{cohpaper}  In both of these methods, the
normalized least mean square (NLMS) algorithm is engaged
\cite{Haykin96}  The $m^{\rm th}$ element of the weight vector in
each stage is the true transmitted binary value of the $m^{\rm th}$
user divided by its hard estimate value from the previous stage  The
magnitude of all weight elements in all stages are equal to unity 
Unlike the LMS-PPIC, the PLMS-PPIC method tries to keep this
property in each iteration by using a set of NLMS algorithms with
different step-sizes instead of one NLMS algorithm used in LMS-PPIC 
In each iteration, the parameter estimate of the NLMS algorithm is
chosen whose element magnitudes of cancelation weight estimate have
the best match with unity  In PLMS-PPIC implementation it is assumed
that the receiver knows the phases of all user channels  However in
practice, these phases are not known and should be estimated  In
this paper we improve the PLMS-PPIC procedure \cite{cohpaper} in
such a way that when there is only a partial information of the
channel phases, this modified version simultaneously estimates the
phases and the cancelation weights  The partial information is the
quarter of each channel phase in $(0,2\pi)$ 

The rest of the paper is organized as follows: In section \ref{S4}
the modified version of PLMS-PPIC with capability of channel phase
estimation is introduced  In section \ref{S5} some simulation
examples illustrate the results of the proposed method  Finally the
paper is concluded in section \ref{S6} 

\section{Multistage Parallel Interference Cancelation: Modified PLMS-PPIC Method}\label{S4}

We assume $M$ users synchronously send their symbols
$\alpha_1,\alpha_2,\cdots,\alpha_M$ via a base-band CDMA
transmission system where $\alpha_m\in\{-1,1\}$  The $m^{th}$ user
has its own code $p_m( )$ of length $N$, where $p_m(n)\in \{-1,1\}$,
for all $n$  It means that for each symbol $N$ bits are transmitted
by each user and the processing gain is equal to $N$  At the
receiver we assume that perfect power control scheme is applied 
Without loss of generality, we also assume that the power gains of
all channels are equal to unity and users' channels do not change
during each symbol transmission (it can change from one symbol
transmission to the next one) and the channel phase $\phi_m$ of
$m^{th}$ user is unknown for all $m=1,2,\cdots,M$ (see
\cite{cohpaper} for coherent transmission)  According to the above
assumptions the received signal is
\begin{equation}
\label{e1} r(n)=\sum\limits_{m=1}^{M}\alpha_m
e^{j\phi_m}p_m(n)+v(n),~~~~n=1,2,\cdots,N,
\end{equation}
where $v(n)$ is the additive white Gaussian noise with zero mean and
variance $\sigma^2$  Multistage parallel interference cancelation
method uses $\alpha^{s-1}_1,\alpha^{s-1}_2,\cdots,\alpha^{s-1}_M$,
the bit estimates outputs of the previous stage, $s-1$, to estimate
the related MAI of each user  It then subtracts it from the received
signal $r(n)$ and makes a new decision on each user variable
individually to make a new variable set
$\alpha^{s}_1,\alpha^{s}_2,\cdots,\alpha^{s}_M$ for the current
stage $s$  Usually the variable set of the first stage (stage $0$)
is the output of a conventional detector  The output of the last
stage is considered as the final estimate of transmitted bits  In
the following we explain the structure of a modified version of the
PLMS-PIC method \cite{cohpaper} with simultaneous capability of
estimating the cancelation weights and the channel phases 

Assume $\alpha_m^{(s-1)}\in\{-1,1\}$ is a given estimate of
$\alpha_m$ from stage $s-1$  Define
\begin{equation}
\label{e6} w^s_{m}=\frac{\alpha_m}{\alpha_m^{(s-1)}}e^{j\phi_m} 
\end{equation}
From (\ref{e1}) and (\ref{e6}) we have
\begin{equation}
\label{e7} r(n)=\sum\limits_{m=1}^{M}w^s_m\alpha^{(s-1)}_m
p_m(n)+v(n) 
\end{equation}
Define
\begin{subequations}
\begin{eqnarray}
\label{e8} W^s&=&[w^s_{1},w^s_{2},\cdots,w^s_{M}]^T,\\
\label{e9}
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!X^{s}(n)\!\!\!&=&\!\!\![\alpha^{(s-1)}_1p_1(n),\alpha^{(s-1)}_2p_2(n),\cdots,\alpha^{(s-1)}_Mp_M(n)]^T 
\end{eqnarray}
\end{subequations}
where $T$ stands for transposition  From equations (\ref{e7}),
(\ref{e8}) and (\ref{e9}), we have
\begin{equation}
\label{e10} r(n)=W^{s^T}X^{s}(n)+v(n) 
\end{equation}
Given the observations $\{r(n),X^{s}(n)\}^{N}_{n=1}$, in modified
PLMS-PPIC, like the PLMS-PPIC \cite{cohpaper}, a set of NLMS
adaptive algorithm are used to compute
\begin{equation}
\label{te1} W^{s}(N)=[w^{s}_1(N),w^{s}_2(N),\cdots,w^{s}_M(N)]^T,
\end{equation}
which is an estimate of $W^s$ after iteration $N$  To do so, from
(\ref{e6}), we have
\begin{equation}
\label{e13} |w^s_{m}|=1 ~~~m=1,2,\cdots,M,
\end{equation}
which is equivalent to
\begin{equation}
\label{e14} \sum\limits_{m=1}^{M}||w^s_{m}|-1|=0 
\end{equation}
We divide $\Psi=\left(0,1-\sqrt{\frac{M-1}{M}}\right]$, a sharp
range for $\mu$ (the step-size of the NLMS algorithm) given in
\cite{sg2005}, into $L$ subintervals and consider $L$ individual
step-sizes $\Theta=\{\mu_1,\mu_2,\cdots,\mu_L\}$, where
$\mu_1=\frac{1-\sqrt{\frac{M-1}{M}}}{L}, \mu_2=2\mu_1,\cdots$, and
$\mu_L=L\mu_1$  In each stage, $L$ individual NLMS algorithms are
executed ($\mu_l$ is the step-size of the $l^{th}$ algorithm)  In
stage $s$ and at iteration $n$, if
$W^{s}_k(n)=[w^s_{1,k},\cdots,w^s_{M,k}]^T$, the parameter estimate
of the $k^{\rm th}$ algorithm, minimizes our criteria, then it is
considered as the parameter estimate at time iteration $n$  In other
words if the next equation holds
\begin{equation}
\label{e17} W^s_k(n)=\arg\min\limits_{W^s_l(n)\in I_{W^s}
}\left\{\sum\limits_{m=1}^{M}||w^s_{m,l}(n)|-1|\right\},
\end{equation}
where $W^{s}_l(n)=W^{s}(n-1)+\mu_l \frac{X^s(n)}{\|X^s(n)\|^2}e(n),
~~~ l=1,2,\cdots,k,\cdots,L-1,L$ and
$I_{W^s}=\{W^s_1(n),\cdots,W^s_L(n)\}$, then we have
$W^s(n)=W^s_k(n)$, and therefore all other algorithms replace their
weight estimate by $W^{s}_k(n)$  At time instant $n=N$, this
procedure gives $W^s(N)$, the final estimate of $W^s$, as the true
parameter of stage $s$ 

Now consider $R=(0,2\pi)$ and divide it into four equal parts
$R_1=(0,\frac{\pi}{2})$, $R_2=(\frac{\pi}{2},\pi)$,
$R_3=(\pi,\frac{3\pi}{2})$ and $R_4=(\frac{3\pi}{2},2\pi)$  The
partial information of channel phases (given by the receiver) is in
a way that it shows each $\phi_m$ ($m=1,2,\cdots,M$) belongs to
which one of the four quarters $R_i,~i=1,2,3,4$  Assume
$W^{s}(N)=[w^{s}_1(N),w^{s}_2(N),\cdots,w^{s}_M(N)]^T$ is the weight
estimate of the modified algorithm PLMS-PPIC at time instant $N$ of
the stage $s$  From equation (\ref{e6}) we have
\begin{equation}
\label{tt3}
\phi_m=\angle({\frac{\alpha^{(s-1)}_m}{\alpha_m}w^s_m}) 
\end{equation}
We estimate $\phi_m$ by $\hat{\phi}^s_m$, where
\begin{equation}
\label{ee3}
\hat{\phi}^s_m=\angle{(\frac{\alpha^{(s-1)}_m}{\alpha_m}w^s_m(N))} 
\end{equation}
Because $\frac{\alpha^{(s-1)}_m}{\alpha_m}=1$ or $-1$, we have
\begin{eqnarray}
\hat{\phi}^s_m=\left\{\begin{array}{ll} \angle{w^s_m(N)} &
\mbox{if}~
\frac{\alpha^{(s-1)}_m}{\alpha_m}=1\\
\pm\pi+\angle{w^s_m(N)} & \mbox{if}~
\frac{\alpha^{(s-1)}_m}{\alpha_m}=-1\end{array}\right 
\end{eqnarray}
Hence $\hat{\phi}^s_m\in P^s=\{\angle{w^s_m(N)},
\angle{w^s_m(N)+\pi, \angle{w^s_m(N)}-\pi}\}$  If $w^s_m(N)$
sufficiently converges to its true value $w^s_m$, the same region
for $\hat{\phi}^s_m$ and $\phi_m$ is expected  In this case only one
of the three members of $P^s$ has the same region as $\phi_m$  For
example if $\phi_m \in (0,\frac{\pi}{2})$, then $\hat{\phi}^s_m \in
(0,\frac{\pi}{2})$ and therefore only $\angle{w^s_m(N)}$ or
$\angle{w^s_m(N)}+\pi$ or $\angle{w^s_m(N)}-\pi$ belongs to
$(0,\frac{\pi}{2})$  If, for example, $\angle{w^s_m(N)}+\pi$ is such
a member between all three members of $P^s$, it is the best
candidate for phase estimation  In other words,
\[\phi_m\approx\hat{\phi}^s_m=\angle{w^s_m(N)}+\pi \]
We admit that when there is a member of $P^s$ in the quarter of
$\phi_m$, then $w^s_m(N)$ converges  What would happen when non of
the members of $P^s$ has the same quarter as $\phi_m$? This
situation will happen when the absolute difference between $\angle
w^s_m(N)$ and $\phi_m$ is greater than $\pi$  It means that
$w^s_m(N)$ has not converged yet  In this case where we can not
count on $w^s_m(N)$, the expected value is the optimum choice for
the channel phase estimation, e g  if $\phi_m \in (0,\frac{\pi}{2})$
then $\frac{\pi}{4}$ is the estimation of the channel phase
$\phi_m$, or if $\phi_m \in (\frac{\pi}{2},\pi)$ then
$\frac{3\pi}{4}$ is the estimation of the channel phase $\phi_m$ 
The results of the above discussion are summarized in the next
equation
\begin{eqnarray}
\nonumber \hat{\phi}^s_m = \left\{\begin{array}{llll} \angle
{w^s_m(N)} & \mbox{if}~
\angle{w^s_m(N)}, \phi_m\in R_i,~~i=1,2,3,4\\
\angle{w^s_m(N)}+\pi & \mbox{if}~ \angle{w^s_m(N)}+\pi, \phi_m\in
R_i,~~i=1,2,3,4\\
\angle{w^n_m(N)}-\pi & \mbox{if}~ \angle{w^s_m(N)}-\pi, \phi_m\in
R_i,~~i=1,2,3,4\\
\frac{(i-1)\pi+i\pi}{4} & \mbox{if}~ \phi_m\in
R_i,~~\angle{w^s_m(N)},\angle
{w^s_m(N)}\pm\pi\notin R_i,~~i=1,2,3,4 \\
\end{array}\right 
\end{eqnarray}
Having an estimation of the channel phases, the rest of the proposed
method is given by estimating $\alpha^{s}_m$ as follows:
\begin{equation}
\label{tt4}
\alpha^{s}_m=\mbox{sign}\left\{\mbox{real}\left\{\sum\limits_{n=1}^{N}
q^s_m(n)e^{-j\hat{\phi}^s_m}p_m(n)\right\}\right\},
\end{equation}
where
\begin{equation} \label{tt5}
q^{s}_{m}(n)=r(n)-\sum\limits_{m^{'}=1,m^{'}\ne
m}^{M}w^{s}_{m^{'}}(N)\alpha^{(s-1)}_{m^{'}} p_{m^{'}}(n) 
\end{equation}
The inputs of the first stage $\{\alpha^{0}_m\}_{m=1}^M$ (needed for
computing $X^1(n)$) are given by
\begin{equation}
\label{qte5}
\alpha^{0}_m=\mbox{sign}\left\{\mbox{real}\left\{\sum\limits_{n=1}^{N}
r(n)e^{-j\hat{\phi}^0_m}p_m(n)\right\}\right\} 
\end{equation}
Assuming $\phi_m\in R_i$, then
\begin{equation}
\label{qqpp} \hat{\phi}^0_m =\frac{(i-1)\pi+i\pi}{4} 
\end{equation}
Table \ref{tab4} shows the structure of the modified PLMS-PPIC
method  It is to be notified that
\begin{itemize}
\item Equation (\ref{qte5}) shows the conventional bit detection
method when the receiver only knows the quarter of channel phase in
$(0,2\pi)$  \item With $L=1$ (i e  only one NLMS algorithm), the
modified PLMS-PPIC can be thought as a modified version of the
LMS-PPIC method 
\end{itemize}

In the following section some examples are given to illustrate the
effectiveness of the proposed method 

\section{Simulations}\label{S5}

In this section we have considered some simulation examples 
Examples \ref{ex2}-\ref{ex4} compare the conventional, the modified
LMS-PPIC and the modified PLMS-PPIC methods in three cases: balanced
channels, unbalanced channels and time varying channels  In all
examples, the receivers have only the quarter of each channel phase 
Example \ref{ex2} is given to compare the modified LMS-PPIC and the
PLMS-PPIC in the case of balanced channels 

\begin{example}{\it Balanced channels}:
\label{ex2}
\begin{table}
\caption{Channel phase estimate of the first user (example
\ref{ex2})} \label{tabex5} \centerline{{
\begin{tabular}{|c|c|c|c|c|}
\hline
\multirow{6}{*}{\rotatebox{90}{$\phi_m=\frac{3\pi}{8},M=15~~$}} & N(Iteration) & Stage Number& NLMS & PNLMS  \\
&&&&\\
\cline{2-5} & \multirow{2}{*}{64}& s = 2 &  $\hat{\phi}^s_m=\frac{3 24\pi}{8}$ & $\hat{\phi}^s_m=\frac{3 18\pi}{8}$ \\
\cline{3-5} & & s = 3 & $\hat{\phi}^s_m=\frac{3 24\pi}{8}$ & $\hat{\phi}^s_m=\frac{3 18\pi}{8}$ \\
\cline{2-5} & \multirow{2}{*}{256}& s = 2 &  $\hat{\phi}^s_m=\frac{2 85\pi}{8}$ & $\hat{\phi}^s_m=\frac{2 88\pi}{8}$ \\
\cline{3-5} & & s = 3 & $\hat{\phi}^s_m=\frac{2 85\pi}{8}$ & $\hat{\phi}^s_m=\frac{2 88\pi}{8}$ \\
\cline{2-5} \hline
\end{tabular} }}
\end{table}
Consider the system model (\ref{e7}) in which $M$ users
synchronously send their bits to the receiver through their
channels  It is assumed that each user's information consists of
codes of length $N$  It is also assumd that the signal to noise
ratio (SNR) is 0dB  In this example there is no power-unbalanced or
channel loss is assumed  The step-size of the NLMS algorithm in
modified LMS-PPIC method is $\mu=0 1(1-\sqrt{\frac{M-1}{M}})$ and
the set of step-sizes of the parallel NLMS algorithms in modified
PLMS-PPIC method are
$\Theta=\{0 01,0 05,0 1,0 2,\cdots,1\}(1-\sqrt{\frac{M-1}{M}})$,
i e  $\mu_1=0 01(1-\sqrt{\frac{M-1}{M}}),\cdots,
\mu_4=0 2(1-\sqrt{\frac{M-1}{M}}),\cdots,
\mu_{12}=(1-\sqrt{\frac{M-1}{M}})$  Figure~\ref{Figexp1NonCoh}
illustrates the bit error rate (BER) for the case of two stages and
for $N=64$ and $N=256$  Simulations also show that there is no
remarkable difference between results in two stage and three stage
scenarios  Table~\ref{tabex5} compares the average channel phase
estimate of the first user in each stage and over $10$ runs of
modified LMS-PPIC and PLMS-PPIC, when the the number of users is
$M=15$ 
\end{example}

Although LMS-PPIC and PLMS-PPIC, as well as their modified versions,
are structured based on the assumption of no near-far problem
(examples \ref{ex3} and \ref{ex4}), these methods and especially the
second one have remarkable performance in the cases of unbalanced
and/or time varying channels 

\begin{example}{\it Unbalanced channels}:
\label{ex3}
\begin{table}
\caption{Channel phase estimate of the first user (example
\ref{ex3})} \label{tabex6} \centerline{{
\begin{tabular}{|c|c|c|c|c|}
\hline
\multirow{6}{*}{\rotatebox{90}{$\phi_m=\frac{3\pi}{8},M=15~~$}} & N(Iteration) & Stage Number& NLMS & PNLMS  \\
&&&&\\
\cline{2-5} & \multirow{2}{*}{64}& s=2 &  $\hat{\phi}^s_m=\frac{2 45\pi}{8}$ & $\hat{\phi}^s_m=\frac{2 36\pi}{8}$ \\
\cline{3-5} & & s=3 & $\hat{\phi}^s_m=\frac{2 71\pi}{8}$ & $\hat{\phi}^s_m=\frac{2 80\pi}{8}$ \\
\cline{2-5} & \multirow{2}{*}{256}& s=2 &  $\hat{\phi}^s_m=\frac{3 09\pi}{8}$ & $\hat{\phi}^s_m=\frac{2 86\pi}{8}$ \\
\cline{3-5} & & s=3 & $\hat{\phi}^s_m=\frac{2 93\pi}{8}$ & $\hat{\phi}^s_m=\frac{3 01\pi}{8}$ \\
\cline{2-5} \hline
\end{tabular} }}
\end{table}
Consider example \ref{ex2} with power unbalanced and/or channel loss
in transmission system, i e  the true model at stage $s$ is
\begin{equation}
\label{ve7} r(n)=\sum\limits_{m=1}^{M}\beta_m
w^s_m\alpha^{(s-1)}_m c_m(n)+v(n),
\end{equation}
where $0<\beta_m\leq 1$ for all $1\leq m \leq M$  Both the LMS-PPIC
and the PLMS-PPIC methods assume the model (\ref{e7}), and their
estimations are based on observations $\{r(n),X^s(n)\}$, instead of
$\{r(n),\mathbf{G}X^s(n)\}$, where the channel gain matrix is
$\mathbf{G}=\mbox{diag}(\beta_1,\beta_2,\cdots,\beta_m)$  In this
case we repeat example \ref{ex2}  We randomly get each element of
$G$ from $[0,0 3]$  Figure~\ref{Figexp2NonCoh} illustrates the BER
versus the number of users  Table~\ref{tabex6} compares the channel
phase estimate of the first user in each stage and over $10$ runs of
modified LMS-PPIC and modified PLMS-PPIC for $M=15$ 
\end{example}

\begin{example}
\label{ex4} {\it Time varying channels}: Consider example \ref{ex2}
with time varying Rayleigh fading channels  In this case we assume
the maximum Doppler shift of $40$HZ, the three-tap
frequency-selective channel with delay vector of $\{2\times
10^{-6},2 5\times 10^{-6},3\times 10^{-6}\}$sec and gain vector of
$\{-5,-3,-10\}$dB  Figure~\ref{Figexp3NonCoh} shows the average BER
over all users versus $M$ and using two stages 
\end{example}


\section{Conclusion}\label{S6}

In this paper, parallel interference cancelation using adaptive
multistage structure and employing a set of NLMS algorithms with
different step-sizes is proposed, when just the quarter of the
channel phase of each user is known  In fact, the algorithm has been
proposed for coherent transmission with full information on channel
phases in \cite{cohpaper}  This paper is a modification on the
previously proposed algorithm  Simulation results show that the new
method has a remarkable performance for different scenarios
including Rayleigh fading channels even if the channel is
unbalanced 


Context 3:  The original branch line connected Florence, Marion, Canada, Hillsboro, Lehigh, Canton, Galva, McPherson, Conway, Windom, Little River, Mitchell, Lyons, Chase, then connected with the original AT&SF main line at Ellinwood 

In 1887, the Chicago, Kansas and Nebraska Railway extended its main line from Herington to Pratt   This main line connected Herington, Ramona, Tampa, Durham, Waldeck, Canton, Galva, McPherson, Groveland, Inman, Medora, Hutchinson, Whiteside, Partridge, Arlington,  Langdon, Turon, Preston, Natrona, Pratt   In 1888, this main line was extended to Liberal   Later, this line was extended to Tucumcari, New Mexico and Santa Rosa, New Mexico, where it made a connection with the Southern Pacific from El Paso, Texas   The Chicago, Kansas and Nebraska Railway was absorbed by the Chicago, Rock Island and Pacific Railway  This line is also called the "Golden State Route" 

20th century
The National Old Trails Road, also known as the Ocean-to-Ocean Highway, was established in 1912, and was routed through Windom, Conway, McPherson 

Geography

According to the U S  Census Bureau, the county has a total area of , of which  is land and  (0 3%) is water 

Adjacent counties
 Saline County (north)
 Dickinson County (northeast)
 Marion County (east)
 Harvey County (southeast)
 Reno County (southwest)
 Rice County (west)
 Ellsworth County (northwest)

Major highways
  Interstate 135
  U S  Route 56
  U S  Route 81
  K-4
  K-61
  K-153

Demographics

The McPherson Micropolitan Statistical Area includes all of McPherson County 

2000 census
As of the census of 2000, there were 29,554 people, 11,205 households, and 7,966 families residing in the county   The population density was 33 people per square mile (13/km2)   There were 11,830 housing units at an average density of 13 per square mile (5/km2)   The racial makeup of the county was 96 53% White, 0 81% Black or African American, 0 34% Native American, 0 32% Asian, 0 06% Pacific Islander, 0 79% from other races, and 1 16% from two or more races   1 94% of the population were Hispanic or Latino of any race  37 1% were of German, 12 9% Swedish, 12 1% American, 6 7% English and 6 3% Irish ancestry according to Census 2000 

There were 11,205 households, out of which 33 00% had children under the age of 18 living with them, 62 50% were married couples living together, 6 00% had a female householder with no husband present, and 28 90% were non-families  25 50% of all households were made up of individuals, and 11 80% had someone living alone who was 65 years of age or older   The average household size was 2 49 and the average family size was 2 99 

In the county, the population was spread out, with 25



ground_truth: The normalized least mean square (NLMS) algorithm.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: When was McPherson County established as a county?

Read the following text and answer briefly.

Context 3: 40% under the age of 18, 10 30% from 18 to 24, 25 20% from 25 to 44, 21 80% from 45 to 64, and 17 30% who were 65 years of age or older   The median age was 38 years  For every 100 females there were 95 90 males   For every 100 females age 18 and over, there were 92 90 males 

The median income for a household in the county was $41,138, and the median income for a family was $48,243  Males had a median income of $33,530 versus $21,175 for females  The per capita income for the county was $18,921   About 4 20% of families and 6 60% of the population were below the poverty line, including 5 20% of those under age 18 and 8 10% of those age 65 or over 

Government

Presidential elections
McPherson county is often carried by Republican candidates  The last time a Democratic candidate has carried this county was in 1964 by Lyndon B  Johnson 

Laws
Following amendment to the Kansas Constitution in 1986, the county remained a prohibition, or "dry", county until 1996, when voters approved the sale of alcoholic liquor by the individual drink with a 30 percent food sales requirement 

Education

Colleges
 McPherson College in McPherson
 Bethany College in Lindsborg
 Central Christian College in McPherson

Unified school districts
 Smoky Valley USD 400
 McPherson USD 418
 Canton-Galva USD 419
 Moundridge USD 423
 Inman USD 448

School district office in neighboring county
 Goessel USD 411
 Little River-Windom USD 444

Museums
 Birger Sandzén Memorial Gallery in Lindsborg
 McCormick-Deering Days Museum in Inman
 McPherson Museum in McPherson
 Lindsborg Old Mill & Swedish Heritage Museum in Lindsborg
 Kansas Motorcycle Museum in Marquette

Communities

Cities

 Canton
 Galva
 Inman
 Lindsborg
 Marquette
 McPherson (county seat) 
 Moundridge
 Windom

Unincorporated communities
† means a Census-Designated Place (CDP) by the United States Census Bureau 
 Conway
 Elyria†
 Groveland
 Johnstown
 New Gottland
 Roxbury†

Ghost towns
 Alta Mills
 Battle Hill
 Christian
 Doles Park
 Elivon
 King City
 Sweadal

Townships
McPherson County is divided into twenty-five townships   The cities of Lindsborg and McPherson are considered governmentally independent and are excluded from the census figures for the townships   In the following table, the population center is the largest city (or cities) included in that township's population total, if it is of a significant size 

See also
 List of people from McPherson County, Kansas
 National Register of Historic Places listings in McPherson County, Kansas
 McPherson Valley Wetlands
 Maxwell Wildlife Refuge

References

Notes

Further reading

 Wheeler, Wayne Leland  "An Analysis of Social Change in a Swedish-Immigrant Community: The Case of Lindsborg, Kansas " (PhD dissertation, University of Missouri-Columbia; ProQuest Dissertations Publishing, 1959  5905657) 

County
 Through the Years: A Pictorial History of McPherson County; McPherson Sentinel' Heritage House Publishing Co; 1992 
 McPherson County First Courthouse Built About 1869 or 1870; Lindsborg News-Record; March 30, 1959 
 Pioneer Life and Lore of McPherson County, Kansas; Edna Nyquist; Democratic-Opinion Press; 1932 
 A History of the Church of the Brethren in Kansas (includes McPherson College history); Elmer LeRoy Craik; McPherson Daily; Republican Press; 397 pages; 1922 
 Portrait and Biographical Record of Dickinson, Saline, McPherson, and Marion Counties, Kansas; Chapman Bros; 614 pages; 1893 
 Standard Atlas of McPherson County, Kansas; Geo  A  Ogle & Co; 82 pages; 1921 
 Plat Book of McPherson County, Kansas; North West Publishing Co; 50 pages; 1903 
 Edwards' Atlas of McPherson County, Kansas; John P  Edwards; 51 pages; 1884 

Trails
 The Story of the Marking of the Santa Fe Trail by the Daughters of the American Revolution in Kansas and the State of Kansas; Almira Cordry; Crane Co; 164 pages; 1915  (Download 4MB PDF eBook)
 The National Old Trails Road To Southern California, Part 1 (LA to KC); Automobile Club Of Southern California; 64 pages; 1916  (Download 6 8MB PDF eBook)

Mennonite Settlements
 Impact of Mennonite settlement on the cultural landscape of Kansas; Brenda Martin; Kansas State University; 1985/1988  
 Mennonite settlement : the relationship between the physical and cultural environment; Susan Movle; University of Utah; 1975/1886 
 Status of Mennonite women in Kansas in their church and home relationships; Eva Harshbarger; Bluffton College; 1925/1945 

External links

County
 
 McPherson County - Directory of Public Officials
Historical
 , from Hatteberg's People'' on KAKE TV news
Maps
 McPherson County Maps: Current, Historic, KDOT
 Kansas Highway Maps: Current, Historic, KDOT
 Kansas Railroad Maps: Current, 1996, 1915, KDOT and Kansas Historical Society

 
Kansas counties
1867 establishments in Kansas
Populated places established in 1867



ground_truth: McPherson County was established as a county in 1867.