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jian wang edited comment on DATAFU-21 at 4/1/14 1:26 PM: --------------------------------------------------------- Some investigation updates: Based on the theories from paper: http://utopia.duth.gr/~pefraimi/research/data/2007EncOfAlg.pdf, I plan to associate each item with a key X(j) = 1 - pow(U, 1/w(j)), U is a random variable between (0,1). Then follow the thought of Random Sort, we sort the items in ascending order based on X(j) and select the smallest k = p * n items. Also as simple random sampling algorithm, we could also consider the possibility of rejecting items applying Maurer's lemma and accepting items applying Bernstein's lemma. Apply Maurer's lemma: we would like to find 0<q1<1, so that we reject items whose key is greater than q1. let Y(j) = 1 if (X(j) < q1) = 0 otherwise {Y(j), j = 1 to n} are independent random variables. E(Y(j)) = Pr(X(j) < q1) * 1 + Pr(X(j) >= q1) * 0 = Pr(1 - pow( U, 1/w(j) ) < q1) = Pr(1 - q1 < pow( U, 1/w(j) )) = Pr(pow(1 - q1, w(j) ) < U) = 1 - pow( 1 - q1, w(j) ) E(Y(j) ^ 2) = E(Y(j)) = 1 - pow(1 - q1, w(j) ) set Y = sum(Y(j), j = 1 to n), Q1 = sum( pow( 1-q1, w(j) ) , j = 1 to n) E(Y) = sum(E(Y(j))) = n - sum( pow( 1-q1, w(j) ) ) = n - Q1 apply Maurer's lemma with t = (1 - p) * n - sum( pow(1 - q1, w(j) ) ) = (1 - p) * n - Q1, since t > 0, Q1 < (1 - p) * n. Solving the inequality, I get abs( Q1 - (1 - p) * n - log(err) ) >= sqrt( log(err) ^ 2 - 2 * p * n * log(err) ) Further assume Q1 < (1 - p) * n + log(err), which also satisfies t > 0, get Q1 <= (1 - p) * n + log(err) - sqrt( log(err) ^ 2 - 2 * p * n * log(err) ) (1) we could get q1 by solving (1) Apply Berstein's lemma: similar to applying Maurer's lemma, we could get a q2 so that we could accept item whose key is smaller than q2, 0 <= q2 <= 1. let Z(j) = 1 if X(j) < q2, = 0 if X(j) >= q2 {Z(j), j = 1 to n} are independent random variables. E(Z(j)) = Pr(X(j) < q2) * 1 + Pr(X(j) >= q2) * 0 = Pr(1 - pow( U, 1/w(j) ) < q2) = 1 - pow(1 - q2, w(j) ) E(Z(j) ^ 2) = E(Z(j)) Z(j) - E(Z(j)) <= 1 - E(Z(j)) = pow(1 - q2, w(j) ) <= 1 = M theta(j) ^ 2 = E(Z(j) ^ 2) - E(Z(j)) ^ 2 <= E(Z(j) ^ 2) = 1 - pow(1 - q2, w(j) ) set Z = sum(Z(j), j = 1 to n) Q2 = sum( pow(1 - q2, w(j) ) ) E(Z) = sum(E(Z(j)), j = 1 to n) = n - sum( pow(1 - q2, w(j) ) ) = n - Q2 apply Berstein's lemma with t = sum( pow(1 - q2, w(j) ) ) - (1 - p) * n = Q2 - (1 - p) * n, I get Q2 >= n * (1 - p) + 2 / 3 * log(err) + 2 / 3 * sqrt( log(err) * (log(err) - 9 * n * p / 2 ) ) (2) we could get q2 by solving (2) Questions: (1) Please help comment on the above approach. Do they overall make sense? (2) I am stuck in getting q1 and q2 by solving (1) and (2) respectively. Would like to seek some advice on it. Some thoughts on how to resolve this, eg: solve (1) Q1 <= (1 - p) * n + log(err) - sqrt( log(err) ^ 2 - 2 * p * n * log(err) ) = F(n, p, err) (1) Q1 = sum( pow( 1-q1, w(j) ) , j = 1 to n), 0 < q1 < 1 Remove the less-than inequality of (1), our target is to get an approximate q1 that makes Q1 close to F(n, p, err), we name it as q1_t. we could observe that: (1) the value of Q1 decreases with the increase of q1. (2) F(n, p, err) >= Q1 >= sum( pow( 1-q1, wmax ) , j = 1 to n) = n * pow( 1 - q1, wmax), wmax is MAX( w(j), j = 1 to n ), we could get a lower bound of q1, q1 >= 1 - pow( F(n, p, err) / n, 1/wmax), this lower bound decreases when wmax and n increases. Then we start from the lower bound and try Newton–Raphson method to approach a better q1 that makes the value of Q1 close to F(n, p, err). After a certain number of iterations, we assign the final value of the predicted q1 to q1_t. The newton code would be like: /* * 1 iteration of Newton–Raphson * The real-valued function is: f(q) = (1 - q) ^ w(0) + (1 - q) ^ w(1) + ... + (1 - q) ^ w(n - 1) - F(n, p, err) * the function's derivative is: f'(q) = -1 * [w(0) * (1 - q) ^ (w(0) - 1) + (1 - q) ^ (w(1) - 1) + ... (1 - q) ^ (w(n - 1) - 1)] * given an initial value of q, calculate a better value of q' = q - f(q) / f'(q) * param q: initial value of q * param F: F(n, p, err) * param weights: {w(j), j = 0 to n -1} */ static double newton(double q, double F, List<Double> weights) { double fq = 0; double fdq = 0; for (Double weight : weights) { fq += Math.pow(1.0 - q, weight); fdq += -1 * Math.pow(1.0 - q, weight - 1.0) * weight; } fq -= F; return q - fq / fdq; } was (Author: king821221): Some investigation updates: Based on the theories from paper: http://utopia.duth.gr/~pefraimi/research/data/2007EncOfAlg.pdf, I plan to associate each item with a key X(j) = 1 - pow(U, 1/w(j)), U is a random variable between (0,1). Then follow the thought of Random Sort, we sort the items in ascending order based on X(j) and select the smallest k = p * n items. Also as simple random sampling algorithm, we could also consider the possibility of rejecting items applying Maurer's lemma and accepting items applying Bernstein's lemma. Apply Maurer's lemma: we would like to find 0<q1<1, so that we reject items whose key is greater than q1. let Y(j) = 1 if (X(j) < q1) = 0 otherwise {Y(j), j = 1 to n} are independent random variables. E(Y(j)) = Pr(X(j) < q1) * 1 + Pr(X(j) >= q1) * 0 = Pr(1 - pow( U, 1/w(j) ) < q1) = Pr(1 - q1 < pow( U, 1/w(j) )) = Pr(pow(1 - q1, w(j) ) < U) = 1 - pow( 1 - q1, w(j) ) E(Y(j) ^ 2) = E(Y(j)) = 1 - pow(1 - q1, w(j) ) set Y = sum(Y(j), j = 1 to n), Q1 = sum( pow( 1-q1, w(j) ) , j = 1 to n) E(Y) = sum(E(Y(j))) = n - sum( pow( 1-q1, w(j) ) ) = n - Q1 apply Maurer's lemma with t = (1 - p) * n - sum( pow(1 - q1, w(j) ) ) = (1 - p) * n - Q1, since t > 0, Q1 < (1 - p) * n. Solving the inequality, I get abs( Q1 - (1 - p) * n - log(err) ) >= sqrt( log(err) ^ 2 - 2 * p * n * log(err) ) Further assume Q1 < (1 - p) * n + log(err), which also satisfies t > 0, get Q1 <= (1 - p) * n + log(err) - sqrt( log(err) ^ 2 - 2 * p * n * log(err) ) (1) we could get q1 by solving (1) Apply Berstein's lemma: similar to applying Maurer's lemma, we could get a q2 so that we could accept item whose key is smaller than q2, 0 <= q2 <= 1. let Z(j) = 1 if X(j) < q2, = 0 if X(j) >= q2 {Z(j), j = 1 to n} are independent random variables. E(Z(j)) = Pr(X(j) < q2) * 1 + Pr(X(j) >= q2) * 0 = Pr(1 - pow( U, 1/w(j) ) < q2) = 1 - pow(1 - q2, w(j) ) E(Z(j) ^ 2) = E(Z(j)) Z(j) - E(Z(j)) <= 1 - E(Z(j)) = pow(1 - q2, w(j) ) <= 1 = M theta(j) ^ 2 = E(Z(j) ^ 2) - E(Z(j)) ^ 2 <= E(Z(j) ^ 2) = 1 - pow(1 - q2, w(j) ) set Z = sum(Z(j), j = 1 to n) Q2 = sum( pow(1 - q2, w(j) ) ) E(Z) = sum(E(Z(j)), j = 1 to n) = n - sum( pow(1 - q2, w(j) ) ) = n - Q2 apply Berstein's lemma with t = sum( pow(1 - q2, w(j) ) ) - (1 - p) * n = Q2 - (1 - p) * n, I get Q2 >= n * (1 - p) + 2 / 3 * log(err) + 2 / 3 * sqrt( log(err) * (log(err) - 9 * n * p / 2 ) ) (2) we could get q2 by solving (2) Questions: (1) Please help comment on the above approach. Do they overall make sense? (2) I am stuck in getting q1 and q2 by solving (1) and (2) respectively. Would like to seek some advice on it. Some thoughts on how to resolve this, eg: solve (1) Q1 <= (1 - p) * n + log(err) - sqrt( log(err) ^ 2 - 2 * p * n * log(err) ) = F(n, p, err) (1) Q1 = sum( pow( 1-q1, w(j) ) , j = 1 to n), 0 < q1 < 1 Remove the less-than inequality of (1), our target is to get an approximate q1 that makes Q1 close to F(n, p, err), we name it as q1_t. we could observe that: (1) the value of Q1 decreases with the increase of q1. (2) F(n, p, err) >= Q1 >= sum( pow( 1-q1, wmax ) , j = 1 to n) = n * pow( 1 - q1, wmax), wmax is MAX( w(j), j = 1 to n ), we could get a lower bound of q1, q1 >= 1 - pow( F(n, p, err) / n, 1/wmax), this lower bound decreases when wmax and n increases. Then we start from the lower bound and try Newton–Raphson method to approach a better q1 that makes the value of Q1 close to F(n, p, err). After a certain number of iterations, we assign the final value of the predicted q1 to q1_t. The newton code would be like: /* * 1 iteration of Newton–Raphson * The real-valued function is: f(q) = (1 - q) ^ w(0) + (1 - q) ^ w(1) + ... + (1 - q) ^ w(n - 1) - F(n, p, err) * the function's derivative is: f'(q) = -1 * [w(0) * (1 - q) ^ (w(0) - 1) + (1 - q) ^ (w(1) - 1) + ... (1 - q) ^ (w(n - 1) - 1)] * given an initial value of q, calculate a better value of q' = q - f(q) / f'(q) * param q: initial value of q * param F: F(n, p, err) * param weights: {w(j), j = 0 to n -1} */ static double newton(double q, double F, List<Double> weights) { double fq = 0; double fdq = 0; for(Double weight : weights) { fq += Math.pow(1.0 - q, weight); fdq += -1 * Math.pow(1.0 - q, weight - 1.0) * weight; } fq -= F; return q - fq / fdq; } > Probability weighted sampling without reservoir > ----------------------------------------------- > > Key: DATAFU-21 > URL: https://issues.apache.org/jira/browse/DATAFU-21 > Project: DataFu > Issue Type: New Feature > Environment: Mac OS, Linux > Reporter: jian wang > Assignee: jian wang > > This issue is used to track investigation on finding a weighted sampler > without using internal reservoir. > At present, the SimpleRandomSample has implemented a good > acceptance-rejection sampling algo on probability random sampling. The > weighted sampler could utilize the simple random sample with slight > modification. > One slight modification is: the present simple random sample generates a > uniform random number lies between (0, 1) as the random variable to accept or > reject an item. The weighted sample may generate this random variable based > on the item's weight and this random number still lies between (0, 1) and > each item's random variable remain independent between each other. > Need further think and experiment the correctness of this solution and how to > implement it in an effective way. -- This message was sent by Atlassian JIRA (v6.2#6252)