# Smoothing Technique

1. Consider the following table where certain keywords from daily Twitter messages of stock brokersand traders are collected, along with whether the stock market (as indicated by the S&P 500 index) went up or down on that day. Each word is represented by a binary feature variable, which is 1 if more than half of all participating brokers or traders used it on that day, and 0 otherwise. The data is given below:
 Twit Buy Sell inflation up inflation down unemployment high unemployment down recession SP500 1 1 1 0 1 0 1 0 UP 2 1 0 1 1 0 1 0 UP 3 1 0 1 0 1 0 0 UP 4 1 0 0 0 0 1 1 UP 5 0 1 0 0 0 1 1 Down 6 0 1 0 1 0 1 1 Down 7 0 1 0 1 0 1 0 Down 8 0 0 1 1 0 0 0 Down 9 1 1 0 0 1 0 1 Down Pr[1|UP] Pr[1|Down] Pr[0|UP] Pr[0|Down]

1a) Fill in the last four rows of the table, giving estimated conditional probabilities of 1 and 0 values in each column given S&P index is up, and given it is down. Also in the last column give the estimated prior probability of the index going up and going down.

1b) A new data comes in for today, containing the words “Buy”, “inflation down”, and “recession”. Compute the estimated probability of market going up and market going down using Naive Bayes method. What will be the prediction for the market for today?

1c) Repeat questions 1a) and 1b), but this time use the Laplace Smoothing technique, with α = 1 and β = 20. Will the prediction for the market behavior change?

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