MySQL query taking too long

Every day or two, I get a warning from my web host saying that a MySQL query is taking a ridiculous amount of time to complete (say, more than 60 seconds). Below is such a query. What can I do to speed it up? I am using W3 Total Cache (so this isn’t happening every minute) and the plugin Event Calendar, among others.

SELECT SQL_CALC_FOUND_ROWS distinct wp_posts.* FROM wp_posts LEFT JOIN wp_term_relationships ON (wp_posts.ID = wp_term_relationships.object_id) LEFT JOIN wp_term_taxonomy ON wp_term_taxonomy.term_taxonomy_id = wp_term_relationships.term_taxonomy_id LEFT JOIN wp_ec3_schedule ec3_sch ON ec3_sch.post_id=id WHERE 1=1 AND wp_posts.ID NOT IN ( SELECT tr.object_id FROM wp_term_relationships AS tr INNER JOIN wp_term_taxonomy AS tt ON tr.term_taxonomy_id = tt.term_taxonomy_id WHERE tt.taxonomy = 'category' AND tt.term_id IN ('1050') ) AND wp_posts.post_type = 'post' AND (wp_posts.post_status = 'publish') AND NOT EXISTS (SELECT * FROM wp_term_relationships JOIN wp_term_taxonomy ON wp_term_taxonomy.term_taxonomy_id = wp_term_relationships.term_taxonomy_id WHERE wp_term_relationships.object_id = wp_posts.ID AND wp_term_taxonomy.taxonomy = 'category' AND wp_term_taxonomy.term_id IN (533,3567) ) AND ec3_sch.post_id IS NULL GROUP BY wp_posts.ID ORDER BY wp_posts.post_date DESC LIMIT 0, 10