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 may happen if the VR headset lenses are exposed to sunlight or strong light?

Read the following text and answer briefly.

Context 1: '用户指南 * User Guide 02 CN 11 EN * 包装内含 使用前注意事项 快速引导 产品部件详情说明 操作说明 02 02 03 06 08 01 
•本产品支持在系统设置中进行瞳距调节 , 调节时请务必注意，最小瞳距可能会碰触鼻梁。当您佩戴头盔后，您 “显示”中进行手动调节，请注意设置使用不合适的瞳距，可能会引起视觉重影或者眼睛疲劳。 可在“设置” ► •本产品“护眼模式”经德国 TÜV Rheinland 低蓝光认证，通过软件算法降低三色通道中的蓝光量，来达到保护 “护眼” “色彩调节” 眼睛的作用，该模式下画面颜色偏黄，您可根据个人喜好在“设置” 中激活或关闭此功能。 ““显示” ► ► ► 包装内含： VR 头盔 / 手柄 × 2 / 1 5V AA 碱性干电池 × 4/ 眼镜支架 / 遮光鼻托 / 手柄挂绳 × 2 / USB-C 电源适配器 / USB-Cto C 2 0 数据线 / 快速指南 / 用户指南 / 安全与质保指南使用前注意事项 •本产品在开阔的室內环境使用体验最佳，建议至少预留 2×2 米 的空间。使用前请确认身体没有不适且周围环 境安全，特别是佩戴头盔在室内行走移动时，要尽量避免发生意外。 •不建议 12 岁及以下儿童使用本产品，建议将头盔、手柄和配件置于儿童够不到的位置，13 岁以上青少年须在 成人监护下使用，以免发生意外。 •本产品无近视调节功能，近视用户请佩戴眼镜使用并尽量避免近视眼镜被头盔的光学镜片磨伤或刮伤。建议在 使用和收纳时注意防护光学镜片，避免尖锐物体划伤镜片，擦拭清洁时请使用柔软的眼镜布，否则可能划伤镜片， 影响视觉效果。 •长时间使用可能引发轻微的昡晕或者眼疲劳，建议使用 30 分钟后适当休息，可通过眼保健操或观看远处物体缓 解眼疲劳。如果您的身体感到任何不适，请立即停止使用。如果不适持续，请咨询医生。 •当头盔镜片被阳光或紫外线照射时（尤其在户外、阳台、窗台及汽车内存放时），可能导致屏幕出现永久性黄斑。 请尽量避免该情况发生，此种屏幕损坏不在产品的质保范围内。 *本产品最终外观及功能以实物为准，部分地区包装内含物品有所差异，本说明仅供参考。 02 CN
六自由度 VR 体验 本产品可以追踪头盔和手柄前、后、左、右、上、下和旋转的运动状态，您在现实中的肢体运动会实时反映在虚 拟世界中。 由于没有任何线缆的束缚，您在虚拟世界自由探索时请确保游玩区域的安全。 1  建议准备一个整洁安全的体验空间：至少 2×2 米；保持房间明亮，避免在只有单色的墙或大面积玻璃、镜子类 反射物以及许多移动画面和物体的空间中使用。 2  撕下 VR 头盔前端摄像头上的保护膜，并佩戴手柄挂绳。 3  根据开机后的画面提示进行游玩区域的设定。 ❶ 安装电池 按箭头方向拔出电池盖侧边的绝缘纸 快速引导 提示：本产品虚拟的安全区提醒功能，不能完全保证您在设定好的游戏区域中的安全，请时刻注意周围的安全情况。 提示：建议使用 1 5V AA 碱性电池。 按照图示拨动电池盖拨钮打开电池盖更换电池。 03 CN
❷ 手柄开机 ❸ 头盔开机 ❹ 佩戴头盔，调节至清晰舒适的位置 首次开机：拔出绝缘纸，手柄自动开机（蓝灯闪烁） 非首次开机：短按手柄 Home 键开机（蓝灯闪烁） 长按头盔电源键 2 秒（蓝灯常亮） 调节旋钮转动绑带，使后脑垫套在头上，微调绑带长度及佩戴位置至视野清晰 04 提示：近视用户请佩戴眼镜或镜片插件使用，本产品不具备近视调节功能。 CN
❺ 微调顶绑带 微调顶绑带使其受力以减少额头压力 ❻ 瞳 距 调 节 在系统设置：“设置” ► “显示”界面中进行瞳距调节，点击“+”或“-”按钮可微调瞳距直至画面清晰 64mm 请勿 强行 掰动镜 筒，以 免造 成损坏 ！ 请注 意设 置使用 不合适 的瞳 距，可 能 会引起 视 觉重影 或 者眼睛 疲 劳。准 确 的瞳距 设 置有助 于 获得清 晰 的图像 并 减少眼睛 疲劳。 05 CN
产品部件详情说明 头盔状态指示灯 蓝灯常亮：开机进行中或工作状态 黄灯常亮：充电中，电量低于 98% 红灯常亮：充电中，电量低于 20% 绿灯常亮：充电完毕，电量大于 98% 或 充满 蓝灯闪烁：关机进行中 红灯闪烁：电量低于 20% 指示灯熄灭：休眠或关机 06 ① 电源键 开机：长按 2 秒 关机：长按 5 秒 复位：长按 10 秒 开机时，短按休眠 ② ③ ④ ⑤ 状态指示灯 贴脸泡棉 音量键 彩色透视摄像头 使用时请勿遮挡 ⑥ ⑦ ⑧ 顶部绑带 可拆卸 绑带旋钮 环境追踪摄像头 使用时请勿遮挡 ⑨ ⑩ ⑪ USB-C 接口 左 / 右喇叭 距离传感器 佩戴头盔后，系统自动唤醒 摘下头盔后，系统自动休眠 ⑫ ⑬ 眼球追踪摄像头 此功能仅 Pro 版支持 使用时请勿遮挡 面部追踪摄像头 此功能仅 Pro 版支持 使用时请勿遮挡 CN
手柄状态指示灯 熄灭：已连接或者关机 蓝灯常亮：固件升级模式 蓝灯闪烁：连接中 红蓝灯交替慢速闪烁：等待配对 ① ② 摇杆 菜单键 ③ Home 键 开机 : 短按关机 : 长按 6 秒退出应用 : 短按屏幕中心校正 : 长按 1 秒④ ⑤ ⑥ ⑦ 状态指示灯 抓握键 截屏键 扳机键 ⑧ ⑨ 电池盒 打开：拨动拨钮，电池盒弹出 安装：按压直至自动锁紧 追踪光环 使用时请勿遮挡 注：手柄挂绳可按图示将粗绳穿过细绳并锁紧在手柄尾端 07 CN
手柄硬件复位 如果手柄出现按 Home 键和任何按键均无反应或者头盔中虚拟手柄卡死不动的问题可拆装电池重新启动手柄。 近视用户配戴 本设备不具备近视调节功能，头盔可支持佩戴镜框宽度小于 150mm 的大多数标准眼镜。 操作说明 头控模式 未连接手柄的情况下，您可通过转动头部光标及点击头盔音量加减按键进行操作。 切换主控手柄射线 在主控菜单下，短按对应手柄的扳机键可以切换主控手柄的射线。 屏幕中心校正 戴着头盔直视前方，按住手柄 Home 键（或头控模式下头盔上的音量减键）1 秒以上，进行屏幕中心的校正将菜 单拉到当前视野朝向位置。 断开手柄 长按手柄 Home 键直至手柄状态指示灯红灯亮起并伴随振动产生时即可松手，此时手柄关机并断开与头盔的连接。 您无需刻意进行手柄关机操作，在以下状态下手柄会自动关机省电： •头盔进入深度休眠时（摘下头盔后一段时间） •头盔手柄管理界面解绑手柄时 •头盔关机时 添加新手柄 如需添加新手柄（头盔最多可同时连接一对手柄，即左右手柄各一只），或解绑手柄后再次连接 , 可进入“设置” “手 柄”，点击“配对”，同时按住手柄 Home 键和扳机键直至手柄状态指示灯红蓝交替闪烁时即可松开，然后根据 头盔画面提示操作。 ► 休眠 / 唤醒 方式一：摘下头盔一段时间后，系统自动休眠；戴上头盔时，系统自动唤醒。 方式二：短按头盔电源键也可以进行休眠或唤醒操作。 硬件复位 头盔硬件复位 如果头盔出现短按头盔电源键没有反应或头盔的画面卡死等问题，可以长按头盔电源键 10 秒以上重新启动头盔。 08 CN
安装眼镜支架 安装遮光鼻托 如您存在眼镜摩擦光学镜片或者压迫鼻梁的问题，请按照图示安装眼镜支架以增加间隔空间。 您可根据佩戴的舒适度选择是否安装。 如您感觉鼻子处漏光影响体验，请按照图示安装遮光鼻托配件。 由于眼睛空间密闭可能加剧起雾及出汗问题，您可根据喜好选择是否安装。 ❶ 摘下贴脸泡棉 ❷ 将眼镜支架按照图示安装在产品上 ❸ 将贴脸泡棉按照图示安装眼镜支架上 ❶ 摘下贴脸泡棉 ❸ 安装贴脸泡棉❷ 将遮光鼻托按照图示方式安装在贴脸泡棉上 注：按照图示拆卸眼镜支架 09 CN
更换贴脸泡棉 贴脸泡棉多次清洁和长时间使用后会变色和质地变软，您可酌情更换新泡棉。 更换顶绑带 摘下贴脸泡棉 ❸ 安装贴脸泡棉 按照图示捏住顶绑带金属扣，往下压到底然后抽出 ❷ •购买优质热门应用 •畅 聊 社 区， 与 众 多 PICO 玩 家 一起探索 VR 世界 •管理设备更便捷 •参与丰富互动活动 •更多精彩内容等你来发现 ❶ 微 信公 众 号：PICO VR抖音：PICO官 方 旗 舰 店哔 哩 哔 哩：PICO-VR官 方微 博：PICO-VR ❶ ❷ 10 CN
In The Box: VR Headset / 2 Controllers / 4 1 5V AA Alkaline Batteries / Glasses Spacer / Nose Pad / 2 Controller Lan- yards / USB-C Power Adapter / USB-C to C 2 0 Data Cable / Quick Guide / User Guide / Safety and WarrantyGuide Important Health & Safety Notes • This product is designed and intended to be used in an open and safe indoor area, free of anytripping or slipping hazards  To avoid accidents, remain conscious to the potential confines ofyour physical area and respect the boundary of your virtual area whenever you see it  Be sure towear the lanyards when using the Controllers  Make sure that there is enough space around yourhead and body (at least 2 meters by 2 meters) to stretch your arms to avoid damage or injury toyourself, others, and your surroundings  • This product is not recommended for children aged 12 and under  It is recommended to keep headsets,controllers and accessories out of the reach of children  Teenagers aged 13 and over must use it underadult supervision to avoid accidents  • This product is designed to accommodate most prescription glasses  Make sure to wear the VR Headsetin a manner in which the VR Headset lenses do not rub or impair your prescription lenses  • Prolonged use may cause dizziness or eye fatigue  It is recommended to take a break every 30 minutes Try relieving your eyestrain by looking at distant objects  If you feel any discomfort, stop using the prod- uct immediately  If the discomfort persists, seek medical advice • Do not expose the optical lenses to direct sunlight or other strong light sources  Exposure to directsunlight may cause permanent yellow spot damage on the screen  Screen damage caused by sunlightexposure or other strong sources of light is not covered by the warranty  • This product supports interpupillary distance (IPD) adjustment in system settings  When adjusting,please be aware that with the minimum IPD, it may touch the bridge of the nose  You can adjust the IPDaccording to your actual interpupillary distance in "Settings"►"Display"  Please note that using inap- propriate IPD may increase the risk of discomfort  • This product has an “Eye Protection Mode”, certified by TÜV Rheinland (Germany), which can protectyour eyes by reducing blue light in the three color channels using software algorithms  The screen ap- pears yellowish in this mode and you can turn this feature on/off in "Settings"►"Display"►"Color"►“- Eye Protection”  • Protect optical lenses during use and storage to prevent damage, such as scratches or exposure tostrong light or direct sunlight  * Product and packaging are updated regularly, and the functions and contents of the standalone headset may be upgraded in the future Therefore, the content, appearance and functionality listed in this manual and product packaging are subject to change and may notreflect the final product  These instructions are for reference only  * Carefully read this user guide before using the product and share this information with any other users, as it contains important safetyinformation  Keep the user guide as reference for the future  11 EN
6 Degrees of Freedom VR The device can track your translational and rotational movements in all directions (up/down, left/right,forward/backward, pitch, roll, and yaw)  Your movements in the real world will be captured and translatedto what you see in the virtual world when using the appropriate content  Ensure a safe environment before you start your VR experience  1  Clear a safe indoor area of at least 2 meters by 2 meters  Keep the room bright, avoid spaces with main- ly single-colored walls, glass, mirrors, moving pictures or other similar objects  2  Remove the protective film that covers the headset front cameras  Wear the lanyards connected to theControllers  3  Set up your environment by following instructions on the VR Headset screen  Install Batteries ❶Pull the tab to remove the insulating paper  Quick Guide 2 m 2m This product can not guarantee your safety with guardian system, you will need to always pay attention to the surrounding safety  * Note: 1 5V AA alkaline batteries should be used Slide the toggle according to arrow direction toopen the battery case  12 EN
Power on the Controller ❷ First Start: The Controller will start automaticallyafter removing the insulating paper  Others: Short press the Home button for 1second until the status indicator flashes blue Power on the VR Headset ❸ Long press the Power button for 2 seconds untilthe status indicator turns blue Wear Your Headset for a Comfortable Fit and View ❹ Adjust the strap dial to turn the strap so that the back of your head rests on the padding  Fine-tune thelength and position of the strap to give a clear view  * Note: You can use this product with prescription glasses or lenses insert  13 EN
Fine-tune the Top Strap ❺ Fine-tune the head strap to reduce pressure on the forehead  Interpupillary Distance (IPD) Adjustment ❻ In System Setting, go to “Setting” ► “Display” to adjust IPD, tap “+” or “-” button to slightly adjust IPDuntil the picture is clear  14 64mm Please note that inappropriate IPD setting may cause ghosting or eyestrain Accurate IPD setting helps you get a clear imaging and ease eyestrain  EN
Product Details VR Headset Status Indicator Legend Blue: Powered on with battery over 20% Yellow: Charging: Battery is less than 98% Red: Charging: Battery is less than 20% Green: Charging: Battery is more than 98% or charge complete Blue flashing: Shutting down Red flashing: Battery is less than 20% Off: Sleeping or Powered off Power Power on: Long press for 2 seconds Power off: Long press for 5 seconds Hardware reset: Long press for 10 seconds Short press to enter sleep or wake up Status Indicator Face Cushion Volume ① ② ③ ④ ⑤ ⑥ ⑦ ⑧ RGB See Through Camera Do not block during use  Top Strap Removable Strap Dial Tracking Cameras Do not block during use  ⑨ ⑩ ⑪ USB-C Interface Left/Right Speaker Proximity Sensor The system wakes upwhen the VR headset isput on, sleeps when VRheadset is taken off  ⑫ ⑬ Eye Tracking Cameras Pro version only  Do not block during use  Face Tracking Camera Pro version only  Do not block during use  15 EN
Controller Status Indicator Legend Off: Connected or Powered off Blue: Firmware updating in progress Blue flashing: Searching for connection Red and blue flashing alternately: Pairing in progress 16 Joystick Menu ③ ① ② Home Power on: Short pressPower off: Long press for 6 secondsReturn home screen: Short pressScreen recentering: Press for 1 secondStatus Indicator Grip Capture Trigger ④ ⑤ ⑥ ⑦ ⑧ ⑨ Battery Case Open: Slide down the toggle andpop up the battery case  Lock: Push the battery case to lock  Tracking Ring Do not block during use  * Note: Pass the Controller Lanyardthrough the string as shown andlock at the end of the Controller EN
Operating Instructions Headset Control Mode If the Controller is not connected, you can interact with the home screen by moving your head to directthe crosshairs over your intended selection and clicking the Volume Up/Down button on the VR Headset  Switch the pointer of the master Controller In the home screen, short press the Trigger of the corresponding Controller to switch the pointer of themaster Controller  Screen re-centering Wear the VR Headset and look straight ahead, press and hold the Home button of the Controller or VRHeadset ( or the Volume Down button of the VR Headset in head control mode) for more than 1 second tore-center the screen  Disconnect the Controller Press and hold the Home button until the status indicator turns red and the Controller vibrates Controllers will automatically shut down to save power in the following cases:When the VR Headset enters deep sleep (a while after the VR Headset is taken off)When the Controller is unpairedWhen the VR Headset is powered off Add a new Controller If you need to add a new Controller (the VR Headset can only connect one left Controller and one rightController) or reconnect with an unpaired Controller  Go to “Settings” ► “Controller”, click on “Pair” Press and hold the Home button and the Trigger of the Controller at the same time until the red and bluelights of the Controller flashing alternately, and then follow the instructions on the VR Headset screen  Sleep / Wake up Option 1 (Proximity Sensor) Take off VR Headset for automatic sleeping: wear the VR Headset for automat- ic waking up  Option 2 (POWER Button) Press the Power button of the VR Headset for manual sleeping or waking up  Hardware reset VR Headset reset If the visual in the VR Headset freezes, or the VR Headset does not respond after short press the Powerbutton, you can press the Power button of the VR Headset for more than 10 seconds to reboot the VRHeadset  Controller reset If the virtual Controller, the Home button or any buttons of the Controller doesn\'t respond, remove andreinstall the battery case to restart the Controller  The VR Headset Adjustment This device has no myopia adjustment function  The VR Headset allows wearing most standard glasseswith a frame width of less than 150mm  to install Glasses Spacer to increase the space  You can install or not according to your situation  17 EN
Install Glasses Spacer Install Nose Pad If you have glasses collision with headset lens or pressure on the bridge of nose, please follow the pictureto install Glasses Spacer to increase the space  You can install or not according to your situation  If you feel light leaking from your nose, please follow the picture to install Nose Pad to block the light You can consider having it installed at your own discretion  Disassemble the Face Cushion  Install the Glasses Spacer on the Headset  ❸ ❶ ❷ Install the Face Cushion on the Glasses Spacer  Disassemble the Face Cushion  Install the Nose Pad on the Face Cushion  ❶ ❷ Install the Face Cushion on the Headset  ❸ * Note: Disassemble the Glasses Spacer 18 EN
Replace Face Cushion The Face Cushion will have the following phenomena such as color change, surface fluff, soft texture afterlong-term use and repeated cleaning  You can replace a new Face Cushion as needed  Replace Top Strap ❶ ❷ Disassemble the Face Cushion  Pinch the metal buckle of the top strap asshown, press it down and pull it out Install the Face Cushion on  ❸ ❷ ❶ • Purchase high-quality and trending apps • Join PICO Community and explore the VR worldwith other PICO players• Manage your device with ease • Engage in diverse and interactive activities • More exciting features waiting for you 19 EN
'



ground_truth: Exposure to sunlight or strong light may cause permanent yellow spot damage on the screen.

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: 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) 



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

Read the following text and answer briefly.

Context 4: 

English acceded to the role of Finance Minister in the continuing wake of the financial crisis  In response to New Zealand's rising debt, English made budget deficit-reduction his main priority  His first budget outlined three focuses in New Zealand's financial recovery: "improving the business environment and removing roadblocks to growth; investment in productive infrastructure; and improving the way government works"  One of his first acts was creating the National Infrastructure Unit, charged with formulating a plan for infrastructure projects and investments  He commissioned a government-wide spending review, with an aim to reducing government expenditure—with the exceptions of a two-year stimulus package and long-term increases on infrastructure spending 

In April 2011, the Opposition criticised English for suggesting that New Zealand businesses could use New Zealand's low wages to help it compete with Australia  The National Government campaigned for re-election in 2011 on its economic record  The Government boasted growth for five consecutive quarters up to mid-2010, totalling 1 6% of real GDP 

Strong growth resulted in a surplus of $473 million for the 2015/16 financial year, projected to rise to $8 5 billion by 2020/21  In his 2016 Economic and Fiscal Update address, English stated that reducing debt and tackling the costs of the 2016 Kaikōura earthquake were higher priorities than reducing rates of tax 

Allowances issue
In 2009, the media, including TVNZ and TV3 revealed that English was receiving about NZ$900 a week as part of a living allowance for ministers, to live in his own NZ$1 2 million Wellington home  At the time, English also received $276,200 in his annual salary as Deputy Prime Minister  It was also revealed other ministers with homes in the capital city were also claiming accommodation allowances  On 3 August 2009, Prime Minister John Key started a review of the housing allowances claimed by cabinet ministers  English subsequently paid back $12,000 and only claimed about $24,000 a year in living allowances  The Auditor-General's office said in September 2009 that they were making "preliminary enquiries" into parliamentary housing expenses in response to a letter of complaint from Progressive party leader Jim Anderton  Two days later English stated that he would no longer take up any housing allowance and had paid back all the allowance he had received since the November 2008 election 

Prime Minister (2016–2017)

John Key resigned on 12 December, and endorsed English as his successor in the resulting leadership election  Following the drop-out of both Judith Collins and Jonathan Coleman from the leadership election, English was sworn in as the 39th Prime Minister of New Zealand on 12 December 2016 

English appointed his first cabinet on 18 December  In a reshuffle, he appointed Steven Joyce to succeed him as Finance Minister, while most ministerial portfolios remained the same 

In February 2017, English did not attend Waitangi Day commemorations at the historic treaty grounds, reportedly in response to the Ngāpuhi iwi's decision to stop the Prime Minister from speaking at the marae  Ngāpuhi have protested the Government's negotiation of the Trans Pacific Partnership Agreement (TPPA), which the iwi believe infringes upon Māori sovereignty, and thus does not adhere to the Treaty of Waitangi  English had been invited to attend in an official capacity; his non-attendance was criticised by a Ngāpuhi elder and Opposition leader Andrew Little 

In his first overseas trip as Prime Minister, English travelled to Europe to discuss trade ties, including a prospective New Zealand–European Union free trade agreement  He first travelled to London on 13 January 2017 to meet British Prime Minister Theresa May  Discussing trade relations, English said the two nations were "natural partners" and would "continue to forge ties" after the UK's withdrawal from the EU  He also arranged to meet with London Mayor Sadiq Khan, Belgian Prime Minister Charles Michel and German Chancellor Angela Merkel   In a meeting with Merkel, English received crucial backing from Germany for a trade deal with the EU  On 16 January, English stated that his government would continue to promote TPPA, despite the United States' decision to withdraw from the agreement  He explained that Southeast Asian countries would now be treated as a priority in negotiations—he also asserted that the United States was ceding influence to China by its rejection of the trade pact 

At a press conference at the Beehive on 1 February 2017, English announced that the 2017 general election would be held on 23 September  The Prime Minister later confirmed that his party would approach ACT, United Future and the Māori Party if confidence and supply agreements were required to form a government following the election  In his second cabinet reshuffle on 24 April, English appointed Gerry Brownlee as his new Foreign Affairs Minister; he also promoted Nikki Kaye to the portfolio of Education Minister, and moved Mark Mitchell into the cabinet to become Defence Minister  The reshuffle was perceived as an election preparation 

On 13 February 2017, English welcomed Australian Prime Minister Malcolm Turnbull to Wellington  The two leaders reaffirmed their shared trade agenda, and discussed changes to the Australian citizenship pathway which will affect permanent residents originating from New Zealand 

On 19 June, it was reported that Todd Barclay, who succeeded English as MP for Clutha-Southland, had clandestinely recorded one of his employee's conversations the previous year, and that John Key's leaders' budget was used to pay a confidential settlement after the employee resigned  English admitted that he had been aware of the illegal recording and the settlement, and thus implicated in the scandal 

During the 2017 National campaign launch, English introduced a $379 million social investment package including digital learning academies for high school students, more resources for mathematics, and boosting support for teaching second languages in schools, and maintaining National Standards in the school curriculum  Prime Minister English also sought to defend National's financial management and economic track record and claimed that the opposition Labour Party would raise taxes  Early opinion polling had forecast a poor showing in the election for the Labour Party, but in early August 37-year-old Jacinda Ardern took over as Labour leader and seemingly energised younger voters 

At the 2017 general election, National won the largest share of the party vote (44 4%) and the largest number of seats (56) in the House Representatives  However, National lacked enough seats to govern alone due to two of the party's support partners, the Māori Party and United Future, losing their parliamentary seats  In response, English stated that the party would be entering into talks to form a coalition with New Zealand First  Following talks with the two largest parties, New Zealand First entered a coalition arrangement with the Labour Party  English was succeeded as prime minister by Jacinda Ardern on 26 October 

Opposition (2017–2018)

Leader of the Opposition
English was re-elected as National Party leader on 24 October 2017  At the time of his re-election, English announced his intention to stay on as leader until the next general election  On 13 February 2018, however, he stood down as National Party leader due to personal reasons, and instructed the party to put into motion the processes to elect a new leader  He also retired from Parliament  English's resignation followed weeks of speculation that he would step aside for a new leader  On 27 February, he was succeeded as party leader by Simon Bridges as the result of the leadership election held that day 

Post-premiership 
In 2018, English joined the board of Australian conglomerate, Wesfarmers  English serves in Chairmanships of Mount Cook Alpine Salmon, Impact Lab Ltd and Manawanui Support Ltd  He is also a director of The Instillery, Centre for Independent Studies and The Todd Corporation Limited, and is a member of the Impact Advisory Group of Macquarie Infrastructure and Real Assets 

Political and social views

English is regarded as more socially conservative than his predecessor, John Key  He has stated his opposition to voluntary euthanasia and physician-assisted suicide, same-sex civil unions, and the decriminalisation of prostitution  As Prime Minister he opposed any "liberalisation" of abortion law 

In 2004, English voted against a bill to establish civil unions for both same-sex and opposite-sex couples  In 2005, he voted for the Marriage (Gender Clarification) Amendment Bill, which would have amended the Marriage Act to define marriage as only between a man and a woman  English voted against the Marriage (Definition of Marriage) Amendment Bill, a bill that legalised same-sex marriage in New Zealand  However, in December 2016 he stated, "I'd probably vote differently now on the gay marriage issue  I don't think that gay marriage is a threat to anyone else's marriage" 

In 2009, English voted against the Misuse of Drugs (Medicinal Cannabis) Amendment Bill, a bill aimed at amending the Misuse of Drugs Act so that cannabis could be used for medical purposes 

Personal life 
English met his future wife, Mary Scanlon, at university  She was studying medicine at the time, and became a general practitioner  Both her parents were immigrants, her father being Samoan and her mother Italian, born on the island of Stromboli  They have six children: a daughter and five sons 

English is a practising Roman Catholic, but has stated that he considers his religious beliefs personal and thus separate from politics 

In June 2002, English took part in TV3's Fight For Life, a celebrity boxing fundraiser to raise money for the Yellow Ribbon anti-youth-suicide campaign, influenced by the death of a teenage nephew in 1997  He lost a split decision to former university colleague Ted Clarke 

Honours
In the 2018 Queen's Birthday Honours, English was appointed a Knight Companion of the New Zealand Order of Merit, for services of over 27 years to the State 

See also

List of New Zealand governments
Politics of New Zealand

References

External links

Profile at National Party 
Profile on Parliament nz
Releases and speeches at Beehive govt nz

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1961 births
21st-century New Zealand politicians
Candidates in the 2017 New Zealand general election
Deputy Prime Ministers of New Zealand
Leaders of the Opposition (New Zealand)
Living people
Members of the Cabinet of New Zealand
Members of the New Zealand House of Representatives
New Zealand farmers
New Zealand finance ministers
New Zealand list MPs
New Zealand MPs for South Island electorates
New Zealand National Party MPs
New Zealand National Party leaders
New Zealand Roman Catholics
New Zealand people of Irish descent
People educated at St  Patrick's College, Silverstream
People from Dipton, New Zealand
People from Lumsden, New Zealand
Prime Ministers of New Zealand
University of Otago alumni
Victoria University of Wellington alumni
Knights Companion of the New Zealand Order of Merit




ground_truth: Power-law functions.