Interview problem: Welford's online algorithm

Problem: 
Provide an online algorithm to compute the sample mean and sample variance of stream data. 





Solution: Let x_1, x_2, \cdots, x_n, \cdots be the stream data, the sample mean \bar{x}_n \equiv\sum_{i=1}^n x_i / n can be computed by \begin{equation}\bar{x}_n=\bar{x}_{n-1} +\frac{x_n - \bar{x}_{n-1}}{n}\,.\tag{1}\end{equation}
For the sample variance s^2_{n}\equiv \frac{1}{n-1}\sum_{i=1}^n\left(x_i-\bar{x}_n\right)^2, a naive method is to compute \begin{equation} s^2_n = \frac{1}{n-1}\left(\sum_{i=1}^n x_i^2 - \frac{\left(\sum_{i=1}^n x_i\right)^2}{n}\right)\end{equation} by the statistics of the sum and sum of squared. However, such method is numerically unstable since it may involve the cancellation of two very similar numbers. Instead, as suggested in this wiki, we use Welford's algorithm: \begin{eqnarray} M_{2,n} &=& M_{2, n-1}+\left(x_n-\bar{x}_{n-1}\right)\left(x_n-\bar{x}_n\right)\,, \\ s_n^2 &=&\frac{M_{2, n}}{n-1}\,.\tag{2}\end{eqnarray}
Proof: Obviously M_{2,n}=\sum_{i=1}^n\left(x_i-\bar{x}_n\right)^2. Note that \begin{eqnarray}\sum_{i=1}^{n-1} \left(\bar{x}_n-\bar{x}_{n-1}\right)^2&=&(n-1)\bar{x}_n^2+(n-1)\bar{x}^2_{n-1}-2(n-1) \bar{x}_n \bar{x}_{n-1}\\&=&-\bar{x}_n^2 +\bar{x}_n\bar{x}_{n-1} + \bar{x}_n\left[n\bar{x}_n - (n-1)\bar{x}_{n-1}\right] - \bar{x}_{n-1}\left[n\bar{x}_n - (n-1)\bar{x}_{n-1}\right]\\ &=&-\bar{x}_n^2 +\bar{x}_n\bar{x}_{n-1} + \bar{x}_n x_n -\bar{x}_{n-1}x_n\,.\end{eqnarray} As a result, \begin{eqnarray}M_{2,n}&=&\sum_{i=1}^{n-1}\left(x_i-\bar{x}_{n-1} +\bar{x}_{n-1}-\bar{x}_n\right)^2 + \left(x_n-\bar{x}_n\right)^2\\&=&M_{2,n-1} + \sum_{i=1}^{n-1} \left(\bar{x}_n-\bar{x}_{n-1}\right)^2+\left(x_n-\bar{x}_n\right)^2\\&=&M_{2, n-1}+\left(x_n-\bar{x}_{n-1}\right)\left(x_n-\bar{x}_n\right)\,.\end{eqnarray}
Location: 美国加利福尼亚州帕罗奥图

Comments

Post a Comment

Popular posts from this blog

529 Plan

This post is for knowledge sharing only. It is not intended to be investment or tax advice. 529 Plans are tax-advantaged accounts specific for education saving: Contribution are made with after-tax dollars. Growth within the account is tax-free. Withdraw for qualified education expenses is also tax free. Roth IRA First 529 Plans are very similar to Roth IRA . But Roth IRA withdraw under  the 5-year rule  is tax free for any purpose. Therefore, unless for the state tax deduction in some states, it is always better to max out Roth IRA first (through backdoor and mega backdoor  for high income earners) before funding a 529 plan. Note: there is NO California state tax deduction for any 529 plan. 529 or Not? Here are some considerations in reality before funding a 529 plan: What if no qualified education expenses In the worst case that there will be no qualified education expenses, you can still take money out for non-qualified expenses, but you have to pay: federal income ta...
Read more

How to offset W2 tax

This post is for knowledge sharing only. It is not intended to be tax or investment advice. 1. Contribute to pre-tax 401K and HSA As well known, pre-tax 401K and health saving account (HSA) are the only a few tax shelters for W2 workers.  2. Tax-loss harvesting If your total capital losses exceed your capital gains in a tax year, you can deduct up to $3,000 (married filing jointly) from your ordinary income. If your losses are more than $3,000, you can carry over the remaining amount to future years, continuing to offset your income until all the losses are used up. A popular method to generate these deductible losses is  tax-loss harvesting . This involves selling a security at a loss to offset gains or reduce ordinary income, and optionally purchasing a similar, but not "substantially identical", security to avoid missing out on the market. However, be aware of the  wash-sale rule , which disallows claiming a loss if you buy the same or a substantially identical...
Read more

Retirement Accounts

Image
This post is for knowledge sharing only. It is not intended to be investment or tax advice. 0. Account Overview  Table 1 Common retirement  accounts in U.S. I summarizes the common retirement accounts in Table 1:  The second row in Table 1 lists the company-sponsored accounts that are offered through employers. The third row in Table 1 lists the individual retirement accounts (IRA) that is unrelated to employments. The columns in Table 1 are about different tax advantages as described below. WARNING: because of the tax advantages, there are also limitations and specific rules imposed by IRS for early withdrawal before the retirement age. For accounts listed in the second column,  Contributions are made with before-tax dollars . The money taken out will be taxed as regular income in the year of withdrawal. For accounts listed in the third column,  Contributions are made with after-tax dollars . The money taken out is  tax free . For after-tax 401K in the fou...
Read more