# US 10,311,486 B1 · Computer-implemented systems and methods for response curve estimation > Agents: this data is available as JSON from the patentgrep API and MCP server — see https://patentgrep.com/llms.txt - Number: US 10,311,486 B1 - Document ID: US10311486B1 - Status: Granted - Kind: B1 - Application number: 14/275,758 - Filed: 2014-05-12 - Granted: 2019-06-04 - Earliest claimed: 2013-05-13 - Assignees: Oath (Americas) Inc. - Applicants: Oath (Americas) Inc. - Inventors: Niklas Karlsson - Web page: https://patentgrep.com/patent/US10311486B1 ## CPC - G06Q 30/0275: {Auctions} - G06Q: INFORMATION AND COMMUNICATION TECHNOLOGY [ICT] SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES; SYSTEMS OR METHODS SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES, NOT OTHERWISE PROVIDED FOR ## Abstract Systems, methods, and computer-readable media are disclosed for calculating input-output relationships using, for example, spline functions. In accordance with one implementation, a computer-implemented method is provided that includes determining at least one price-volume point comprising a price and associated volume, transforming the price, and using the transformed price to calculate values along a spline function to obtain a price-volume curve weight factor. The method further includes transforming a second price, calculating values along the spline function and the first derivative of the spline function using the transformed second price to obtain a volume estimate and volume rate at the second price, and recalculating the spline function based on the calculated price-volume curve weight factor, volume estimate, volume estimate and volume rate. ## Claims ### Claim 1 A computer-implemented method for an improved online display of electronic content, the method comprising the following operations performed by one or more processors of an estimation system: determining, by the estimation system, an input-output point comprising a first request value and a corresponding first volume value, the estimation system being in communication with a web server and a user device by an electronic network, the estimation system and/or web server being in communication with a request-volume data store; generating, by the estimation system, a first transformed request value by modifying the first request value using a stabilizing factor; generating, by the estimation system, a twice-transformed request value by modifying the first transformed request value based on a control range of request values, the control range comprising a lower boundary and an upper boundary; determining, by the estimation system, values along a spline function to obtain a curve weight factor by recursively calculating a matrix comprising the twice-transformed request value; generating, by the estimation system, a second transformed request value by modifying a second request value using the stabilizing factor, the second request value being different from the first request value; determining, by the estimation system, based on the second transformed request value, values along the spline function and the first derivative of the spline function, to obtain an estimated volume value and estimated volume rate associated with the second request value; determining, by the estimation system, electronic content to display on a web page based on the estimated volume value and the estimated output volume rate; and communicating, from the estimation system to the web server, instructions to display the electronic content on the user device. ### Claim 2 (depends on 1) The method of claim 1, wherein the volume associated with the bid value represents a number of impressions of viewers of the electronic content. ### Claim 3 (depends on 2) The method of claim 2, further comprising: determining a delivery goal associated with the electronic content; determining a number of impressions needed to satisfy the determined delivery goal; and determining, using the spline function, at least one bid value associated with the determined number of impressions. ### Claim 4 (depends on 1) The method of claim 1, wherein the spline function is non-decreasing and is constructed of more than one polynomial function. ### Claim 5 (depends on 1) The method of claim 1, wherein generating a first transformed request value comprises: raising the first request value to the power of the stabilizing factor; and generating a first transformed volume value by determining a logarithm of the first volume value. ### Claim 6 A non-transitory computer-readable storage medium containing instructions that, when executed by at least one electronic processor of an estimation system, causes the at least one electronic processor to perform operations for an improved display of electronic content, the operations comprising: determining, by the estimation system, an input-output point comprising a first request value and a corresponding first volume value, wherein the estimation system is in communication with a web server and a user device by an electronic network, the estimation system and/or web server being in communication with a request-volume data store; generating, by the estimation system, a first transformed request value by modifying the first request value using a stabilizing factor; generating, by the estimation system, a twice-transformed request value by modifying the first transformed request value based on a control range of request values, the control range comprising a lower boundary and an upper boundary; determining, by the estimation system, values along a spline function to obtain a curve weight factor by recursively calculating a matrix comprising the twice-transformed request value; generating, by the estimation system, a second transformed request value by modifying a second request value using the stabilizing factor, the second request value being different from the first request value; determining, by the estimation system, based on the second transformed request value, values along the spline function and the first derivative of the spline function, to obtain an estimated volume value and estimated volume rate associated with the second request value; determining, by the estimation system, electronic content to display on a web page based on the estimated volume value and the estimated volume rate; and communicating, from the estimation system to the web server, instructions to display the electronic content on the user device. ### Claim 7 (depends on 6) The computer-readable medium of claim 6, wherein the volume associated with the bid value represents a number of impressions of viewers of the electronic content. ### Claim 8 (depends on 7) The computer-readable medium of claim 7, wherein the instructions are further configured to cause the at least one processor to: determine a delivery goal associated with the electronic content; determine a number of impressions needed to satisfy the determined delivery goal; and determine, using the spline function, at least one bid value associated with the determined number of impressions. ### Claim 9 (depends on 6) The computer-readable medium of claim 6, wherein the spline function is non-decreasing and is constructed of more than one polynomial functions. ### Claim 10 (depends on 6) The computer-readable medium of claim 6, wherein generating a first transformed request value comprises: raising the first request value to the power of the stabilizing factor; and generating a first transformed volume value by determining a logarithm of the first volume value. ### Claim 11 A system for an improved online display of electronic content, comprising: at least one electronic processor of an estimation system, wherein the estimation system is in communication with a web server and a user device by an electronic network, the estimation system and/or web server being in communication with a request-volume data store; and a storage device that stores instructions that, when executed by the at least one electronic processor, causes the at least one electronic processor to: determine, by the estimation system, an input-output point comprising a first request value and a corresponding first volume value; generate, by the estimation system, a first transformed request value by modifying the first request value using a stabilizing factor; generate, by the estimation system, a twice-transformed request value by modifying the first transformed request value based on a control range of request values, the control range comprising a lower boundary and an upper boundary; determine, by the estimation system, values along a spline function to obtain a curve weight factor by recursively calculating a matrix comprising the twice-transformed request value; generate, by the estimation system, a second transformed request value by modifying a second request value using the stabilizing factor, the second request value being different from the first request value; determine, by the estimation system, based on the second transformed request value, values along the spline function and the first derivative of the spline function, to obtain an estimated volume value and estimated volume rate associated with the second request value; determining, by the estimation system, electronic content to display on a web page based on the estimated volume value and the estimated volume rate, and communicate, from the estimation system to the web server, instructions to display the electronic content on the user device. ### Claim 12 (depends on 11) The system of claim 11, wherein the volume associated with the bid value price represents a number of impressions of viewers of the electronic content. ### Claim 13 (depends on 11) The system of claim 11, wherein the spline function is non-decreasing and is constructed of more than one polynomial functions. ### Claim 14 (depends on 11) The system of claim 11, wherein the first request value is transformed by raising the first request value to the power of the stabilizing factor; and a first transformed volume value is determined by determining a logarithm of the first volume value. ### Claim 15 (depends on 1) The method of claim 1, wherein the input-output point is a first input output point; and recursively calculating a matrix comprises multiplying a base matrix with a forgetting factor, the base matrix being associated with a second input-output point different from the first input-output point. ### Claim 16 (depends on 6) The computer-readable medium of claim 6, wherein the input-output point is a first input output point; and recursively calculating a matrix comprises multiplying a base matrix with a forgetting factor, the base matrix being associated with a second input-output point different from the first input-output point.