In statistics , the explained sum of squares ( ESS ), alternatively known as the model sum of squares or sum of squares due to regression ( SSR – not to be confused with the residual sum of squares (RSS) or sum of squares of errors), is a quantity used in describing how well a model, often a regression model , represents the data being modelled. In particular, the explained sum of squares measures how much variation there is in the modelled values and this is compared to the total sum of squares (TSS), which measures how much variation there is in the observed data, and to the residual sum of squares , which measures the variation in the error between the observed data and modelled values.
31-469: [REDACTED] Look up SSR in Wiktionary, the free dictionary. SSR may refer to: Businesses, entertainment and products [ edit ] Solid State Records , a Christian record label Chevrolet SSR , a Chevrolet small truck Disney's Saratoga Springs Resort & Spa Small Screen Rendering, a technology part of Opera Mini Sonic and
62-740: A New MetroRail project in Perth, Western Australia Springfield Street Railway , a former interurban street railway in Springfield, Massachusetts Strategic steam reserve , a collection of steam locomotives retained for use in a national emergency Sub-surface Railways, cut-and-cover railway lines forming part of the London Underground ; also called Sub-surface Lines Science and technology [ edit ] Scalable Sample Rate, used in MPEG-4 Part 3 and MPEG-2 Part 7 Scalable Source Routing ,
93-525: A New MetroRail project in Perth, Western Australia Springfield Street Railway , a former interurban street railway in Springfield, Massachusetts Strategic steam reserve , a collection of steam locomotives retained for use in a national emergency Sub-surface Railways, cut-and-cover railway lines forming part of the London Underground ; also called Sub-surface Lines Science and technology [ edit ] Scalable Sample Rate, used in MPEG-4 Part 3 and MPEG-2 Part 7 Scalable Source Routing ,
124-420: A diesel-electric radar picket submarine Sustained silent reading , a form of school-based recreational reading Self Supporting Run-flat tires, a type of run-flat tire Svenska Scoutrådet , the national Scouting and Guiding federation of Sweden In air travel passenger name records , a Special Service Request Same-sex relationship Short-sale restriction, a modified uptick rule instituted by
155-420: A diesel-electric radar picket submarine Sustained silent reading , a form of school-based recreational reading Self Supporting Run-flat tires, a type of run-flat tire Svenska Scoutrådet , the national Scouting and Guiding federation of Sweden In air travel passenger name records , a Special Service Request Same-sex relationship Short-sale restriction, a modified uptick rule instituted by
186-494: A limited degree of sequence homology Slope stability radar , the application for the monitoring of slope stability at open-cut mines Solid-state relay , an electronic switching device in which a small control signal controls a larger load current or voltage Sources, sinks and reservoirs , for greenhouse gases Stable salt reactor , a proposed low-cost nuclear reactor design Sum of squared residuals Sum of squares due to regression Server-side rendering , using
217-494: A limited degree of sequence homology Slope stability radar , the application for the monitoring of slope stability at open-cut mines Solid-state relay , an electronic switching device in which a small control signal controls a larger load current or voltage Sources, sinks and reservoirs , for greenhouse gases Stable salt reactor , a proposed low-cost nuclear reactor design Sum of squared residuals Sum of squares due to regression Server-side rendering , using
248-439: A routing algorithm for unstructured networks Screen Space Reflections used in the computer graphics to compute approximate mirror reflections Secondary surveillance radar , a radar system used in air traffic control Simple Sequence Repeat , repeating sequences of 2-6 base pairs of DNA Site-specific recombination , a type of genetic recombination in which DNA strand exchange takes place between segments possessing only
279-439: A routing algorithm for unstructured networks Screen Space Reflections used in the computer graphics to compute approximate mirror reflections Secondary surveillance radar , a radar system used in air traffic control Simple Sequence Repeat , repeating sequences of 2-6 base pairs of DNA Site-specific recombination , a type of genetic recombination in which DNA strand exchange takes place between segments possessing only
310-465: A senior secondary school in India Security sector reform , a process to amend a security sector towards good governance The South Saskatchewan Regiment , a Canadian infantry regiment Special Support and Reconnaissance Company , a Danish military unit The Structure of Scientific Revolutions , a 1962 book by Thomas Kuhn The obsolete United States Navy hull classification symbol for
341-408: A senior secondary school in India Security sector reform , a process to amend a security sector towards good governance The South Saskatchewan Regiment , a Canadian infantry regiment Special Support and Reconnaissance Company , a Danish military unit The Structure of Scientific Revolutions , a 1962 book by Thomas Kuhn The obsolete United States Navy hull classification symbol for
SECTION 10
#1732844418932372-706: A technology part of Opera Mini Sonic and the Secret Rings , a 2007 platform video game Stephen's Sausage Roll , a 2016 puzzle video game Sirius Satellite Radio , a satellite radio service operating in North America Strategic Scientific Reserve, a fictional organization in Marvel Comics and precursor to the fictional organization S.H.I.E.L.D. SSR Wheels , a Japanese wheel manufacturer Geographical locations [ edit ] Soviet Socialist Republic, see Republics of
403-576: A web server to deliver a customized HTML file for a user (client) State-space representation , a particular type of a mathematical model of a physical system, used in control engineering and in satellite navigation People [ edit ] S. S. Rajamouli (born 1973), Indian film director S. S. Rajendran (1928–2014), Indian actor Sushant Singh Rajput (1986–2020), Indian actor Singeetam Srinivasa Rao (born 1931), Indian film director Other uses [ edit ] Swiss Broadcasting Corporation (SRG SSR) Sainik School, Rewa ,
434-576: A web server to deliver a customized HTML file for a user (client) State-space representation , a particular type of a mathematical model of a physical system, used in control engineering and in satellite navigation People [ edit ] S. S. Rajamouli (born 1973), Indian film director S. S. Rajendran (1928–2014), Indian actor Sushant Singh Rajput (1986–2020), Indian actor Singeetam Srinivasa Rao (born 1931), Indian film director Other uses [ edit ] Swiss Broadcasting Corporation (SRG SSR) Sainik School, Rewa ,
465-542: Is y m {\displaystyle y_{m}} times the sum of the elements of y ^ {\displaystyle {\hat {y}}} , so the condition is that the sum of the elements of y equals the sum of the elements of y ^ {\displaystyle {\hat {y}}} , or equivalently that the sum of the prediction errors (residuals) y i − y ^ i {\displaystyle y_{i}-{\hat {y}}_{i}}
496-638: Is The residual vector e ^ {\displaystyle {\hat {e}}} is y − X β ^ = y − X ( X T X ) − 1 X T y {\displaystyle y-X{\hat {\beta }}=y-X(X^{T}X)^{-1}X^{T}y} , so the residual sum of squares e ^ T e ^ {\displaystyle {\hat {e}}^{T}{\hat {e}}} is, after simplification, Denote as y ¯ {\displaystyle {\bar {y}}}
527-1132: Is a vector of ones, the first element of this vector X T e ^ {\displaystyle X^{T}{\hat {e}}} is the sum of the residuals and is equal to zero. This proves that the condition holds for the result that TSS = ESS + RSS . In linear algebra terms, we have R S S = ‖ y − y ^ ‖ 2 {\displaystyle RSS=\|y-{\hat {y}}\|^{2}} , T S S = ‖ y − y ¯ ‖ 2 {\displaystyle TSS=\|y-{\bar {y}}\|^{2}} , E S S = ‖ y ^ − y ¯ ‖ 2 {\displaystyle ESS=\|{\hat {y}}-{\bar {y}}\|^{2}} . The proof can be simplified by noting that y ^ T y ^ = y ^ T y {\displaystyle {\hat {y}}^{T}{\hat {y}}={\hat {y}}^{T}y} . The proof
558-481: Is different from Wikidata All article disambiguation pages All disambiguation pages SSR [REDACTED] Look up SSR in Wiktionary, the free dictionary. SSR may refer to: Businesses, entertainment and products [ edit ] Solid State Records , a Christian record label Chevrolet SSR , a Chevrolet small truck Disney's Saratoga Springs Resort & Spa Small Screen Rendering,
589-426: Is different from Wikidata All article disambiguation pages All disambiguation pages Explained sum of squares The explained sum of squares (ESS) is the sum of the squares of the deviations of the predicted values from the mean value of a response variable, in a standard regression model — for example, y i = a + b 1 x 1 i + b 2 x 2 i + ... + ε i , where y i
620-579: Is the i observation of the response variable , x ji is the i observation of the j explanatory variable , a and b j are coefficients , i indexes the observations from 1 to n , and ε i is the i value of the error term . In general, the greater the ESS, the better the estimated model performs. If a ^ {\displaystyle {\hat {a}}} and b ^ i {\displaystyle {\hat {b}}_{i}} are
651-456: Is the regression intercept, is where y is an n × 1 vector of dependent variable observations, each column of the n × k matrix X is a vector of observations on one of the k explanators, β {\displaystyle \beta } is a k × 1 vector of true coefficients, and e is an n × 1 vector of the true underlying errors. The ordinary least squares estimator for β {\displaystyle \beta }
SECTION 20
#1732844418932682-413: Is zero. This can be seen to be true by noting the well-known OLS property that the k × 1 vector X T e ^ = X T [ I − X ( X T X ) − 1 X T ] y = 0 {\displaystyle X^{T}{\hat {e}}=X^{T}[I-X(X^{T}X)^{-1}X^{T}]y=0} : since the first column of X
713-602: The U.S. Securities and Exchange Commission (SEC) in 2010 Somali State Resistance , a rebel group in Ethiopia Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title SSR . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=SSR&oldid=1219748492 " Category : Disambiguation pages Hidden categories: Short description
744-545: The U.S. Securities and Exchange Commission (SEC) in 2010 Somali State Resistance , a rebel group in Ethiopia Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title SSR . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=SSR&oldid=1219748492 " Category : Disambiguation pages Hidden categories: Short description
775-462: The Secret Rings , a 2007 platform video game Stephen's Sausage Roll , a 2016 puzzle video game Sirius Satellite Radio , a satellite radio service operating in North America Strategic Scientific Reserve, a fictional organization in Marvel Comics and precursor to the fictional organization S.H.I.E.L.D. SSR Wheels , a Japanese wheel manufacturer Geographical locations [ edit ] Soviet Socialist Republic, see Republics of
806-700: The Soviet Union Slovak Soviet Republic (1919), a very short-lived communist state in south and eastern Slovakia Slovak Socialist Republic , the official name of Slovakia from 1969 to 1990 Second Spanish Republic Railways [ edit ] Sabah State Railway , a railway in Malaysia Southern Shorthaul Railroad , an Australian provider of motive power hauling trains in New South Wales and Victoria, and freight train operator Southern Suburbs Railway,
837-485: The Soviet Union Slovak Soviet Republic (1919), a very short-lived communist state in south and eastern Slovakia Slovak Socialist Republic , the official name of Slovakia from 1969 to 1990 Second Spanish Republic Railways [ edit ] Sabah State Railway , a railway in Malaysia Southern Shorthaul Railroad , an Australian provider of motive power hauling trains in New South Wales and Victoria, and freight train operator Southern Suburbs Railway,
868-809: The constant vector all of whose elements are the sample mean y m {\displaystyle y_{m}} of the dependent variable values in the vector y . Then the total sum of squares is The explained sum of squares, defined as the sum of squared deviations of the predicted values from the observed mean of y , is Using y ^ = X β ^ {\displaystyle {\hat {y}}=X{\hat {\beta }}} in this, and simplifying to obtain y ^ T y ^ = y T X ( X T X ) − 1 X T y {\displaystyle {\hat {y}}^{T}{\hat {y}}=y^{T}X(X^{T}X)^{-1}X^{T}y} , gives
899-412: The estimated coefficients , then is the i predicted value of the response variable. The ESS is then: In some cases (see below): total sum of squares (TSS) = explained sum of squares (ESS) + residual sum of squares (RSS). The following equality, stating that the total sum of squares (TSS) equals the residual sum of squares (=SSE : the sum of squared errors of prediction) plus
930-421: The explained sum of squares (SSR :the sum of squares due to regression or explained sum of squares), is generally true in simple linear regression: Square both sides and sum over all i : Here is how the last term above is zero from simple linear regression So, Therefore, The general regression model with n observations and k explanators, the first of which is a constant unit vector whose coefficient
961-405: The result that TSS = ESS + RSS if and only if y T y ¯ = y ^ T y ¯ {\displaystyle y^{T}{\bar {y}}={\hat {y}}^{T}{\bar {y}}} . The left side of this is y m {\displaystyle y_{m}} times the sum of the elements of y , and the right side