TF ErrorAna StatInference - Revision history

Foretony: /* p-Value */

2020-04-12T20:02:31Z

p-Value

Brenleyt: /* p-Value */

2017-04-28T20:28:29Z

p-Value

Brenleyt: /* p-Value */

2017-04-28T20:27:23Z

p-Value

Foretony: /* Least Squares fit to a Polynomial */

2016-04-01T20:22:49Z

Least Squares fit to a Polynomial

Foretony: /* Bayesian inference. */

2016-02-26T19:46:26Z

Bayesian inference.

Foretony: /* Error Matrix */

2015-03-20T17:22:35Z

Error Matrix

Foretony: /* Correlation Probability */

2014-04-04T17:53:22Z

Correlation Probability

Foretony: /* Correlation Probability */

2014-04-01T02:08:06Z

Correlation Probability

Stocjas2: /* The Method of Determinants */

2014-03-21T19:19:00Z

The Method of Determinants

Stocjas2: /* The Method of Determinants */

2014-03-21T19:18:47Z

The Method of Determinants

@@ Line 736: / Line 736: @@
 or
-   TMath::Prob(<math>\chi^2</math>,<math>\nu</math>)
+   TMath::Prob(chi^2,nu)
 You can turn things around the other way by defining P-value to be

@@ Line 709: / Line 709: @@
 The p-value is defined as the cumulative <math>\chi^2</math> probability distribution
-: p-value= <math>P(\chi^2,\nu) = \int_{\chi^2}^{\infty}\frac{x^{\frac{\nu}{2}-1} e^{-\frac{x}{2}}}{2^{\nu/2} \Gamma(\nu/2)} dx</math>
+: p-value= <math>P(\chi^2,\nu) = \int_{\chi^2}^{\infty}\frac{x^{\frac{\nu}{2}-1} e^{-\frac{x^2}{2}}}{2^{\nu/2} \Gamma(\nu/2)} dx</math>
@@ Line 732: / Line 732: @@
   double ROOT::Math::chisquared_cdf_c (double <math>\chi^2</math>, double <math>\nu</math>, double x0 = 0 )
-:<math>=\int_{\chi^2}^{\infty}\frac{x^{\frac{\nu}{2}-1} e^{-\frac{x}{2}}}{2^{\nu/2} \Gamma(\nu/2)} dx</math>
+:<math>=\int_{\chi^2}^{\infty}\frac{x^{\frac{\nu}{2}-1} e^{-\frac{x^2}{2}}}{2^{\nu/2} \Gamma(\nu/2)} dx</math>
 or

← Older revision		Revision as of 20:27, 28 April 2017
Line 741:		Line 741:


−	: P-value= <math>P(\chi^2,\nu) = \int_{\infty}^{\chi^2}\frac{x^{\frac{\nu}{2}-1} e^{-\frac{x}{2}}}{2^{\nu/2} \Gamma(\nu/2)} dx = 1-(p-value)</math>	+	: P-value= <math>P(\chi^2,\nu) = \int_{\infty}^{\chi^2}\frac{x^{\frac{\nu}{2}-1} e^{-\frac{x^2}{2}}}{2^{\nu/2} \Gamma(\nu/2)} dx = 1-(p-value)</math>

	here P-value would be the probability, under the assumption of a hypothesis H , of obtaining data at least as '''compatible''' with H as the data actually observed.		here P-value would be the probability, under the assumption of a hypothesis H , of obtaining data at least as '''compatible''' with H as the data actually observed.

@@ Line 346: / Line 346: @@
 Let's assume we wish to now fit a polynomial instead of a straight line to the data.
-:<math>y(x) = \sum_{j=0}^{n} a_j x^{n}=\sum_{j=0}^{n} a_j f_j(x)</math>
+:<math>y(x) = \sum_{j=0}^{n} a_j x^{j}=\sum_{j=0}^{n} a_j f_j(x)</math>
 :<math>f_j(x) =</math> a function which does not depend on <math>a_j</math>

@@ Line 21: / Line 21: @@
 :Statistical inference is made by using evidence or observations  to update or to newly infer the probability that a hypothesis may be true. The name "Bayesian" comes from the frequent use of Bayes' theorem in the inference process.
-Bayes' theorem relates the conditional probability, conditional and marginal probability, marginal probabilities of events ''A'' and ''B'', where ''B'' has a non-vanishing probability:
+Bayes' theorem relates the conditional probability as well as the conditional and marginal probability of events ''A'' and ''B''  where ''B'' has a non-vanishing probability:
 :<math>P(A|B) = \frac{P(B | A)\, P(A)}{P(B)}\,\! </math>.

← Older revision		Revision as of 17:22, 20 March 2015
Line 532:		Line 532:

	:<math>\frac{\partial a_k}{\partial y_i} = </math>?		:<math>\frac{\partial a_k}{\partial y_i} = </math>?
		+
		+
		+	The parameters were given by our solution to the matrix problem
		+
		+	The parameters <math>a_j</math> in the previous section are linear parameters to a general function which may be a polynomial.
		+
		+	The system of equations is composed of <math>n</math> equations where the <math>k^{\mbox{th}}</math> equation is given as
		+
		+	: <math>\sum_i^N y_i \frac{ f_k(x_i)}{\sigma_i^2} = \sum_{j=0}^{n} a_j \sum_i^N \frac{ f_k(x_i)}{\sigma_i^2} f_j(x_i)</math>
		+
		+
		+	may be represented in matrix form as
		+
		+	: <math>\tilde{\beta} = \tilde{a} \tilde{\alpha}</math>
		+
		+	where
		+	:<math>\beta_k= \sum_i^N y_i \frac{ f_k(x_i)}{\sigma_i^2} </math>
		+
		+	: <math>\alpha_{kj} = \sum_i^N \frac{ f_k(x_i)}{\sigma_i^2} f_j(x_i)</math>
		+
		+	or in matrix form
		+
		+
		+	:<math>\tilde{\beta}= ( \beta_1, \beta_2, \cdots , \beta_n) </math> = a row matrix of order <math>n</math>
		+	:<math>\tilde{a} =( a_1, a_2, \cdots , a_n)</math> = a row matrix of the parameters
		+
		+
		+	: <math>\tilde{\alpha} = \left ( \begin{matrix} \alpha_{11} & \alpha_{12} & \cdots & \alpha_{1j} \\ \alpha_{21} & \alpha_{22}&\cdots&\alpha_{2j} \\ \vdots &\vdots &\ddots &\vdots \\ \alpha_{k1} &\alpha_{k2} &\cdots &\alpha_{kj}\end{matrix} \right )</math> = a <math>k \times j = n \times n</math> matrix
		+
		+	; the objective is to find the parameters <math>a_j</math>
		+	: To find <math>a_j</math> just invert the matrix
		+
		+	: <math>\tilde{\beta} = \tilde{a} \tilde{\alpha}</math>
		+	: <math>\left ( \tilde{\beta} = \tilde{a} \tilde{\alpha} \right) \tilde{\alpha}^{-1}</math>
		+	: <math>\tilde{\beta}\tilde{\alpha}^{-1} = \tilde{a} \tilde{\alpha}\tilde{\alpha}^{-1}</math>
		+	: <math>\tilde{\beta}\tilde{\alpha}^{-1} = \tilde{a} \tilde{1}</math>
		+
		+
		+	: <math> \Rightarrow \tilde{a} =\tilde{\beta}\tilde{\alpha}^{-1} </math>
		+
		+	; Thus if you invert the matrix <math>\tilde{\alpha}</math> you find <math>\tilde{\alpha}^{-1}</math> and as a result the parameters <math>a_j</math>.
		+

← Older revision		Revision as of 17:53, 4 April 2014
Line 328:		Line 328:
	Instead one assumes a correlation of <math>\rho=0</math> in the parent distribution and then compares the sample value of <math>R</math> with what you would get if there were no correlation.		Instead one assumes a correlation of <math>\rho=0</math> in the parent distribution and then compares the sample value of <math>R</math> with what you would get if there were no correlation.

−	The smaller <math>R</math> is the more likely that the data are un-correlated and that the linear fit is ~~correct~~.	+	The smaller <math>R</math> is the more likely that the data are un-correlated and that the linear fit model is not describing the data.

← Older revision		Revision as of 02:08, 1 April 2014
Line 328:		Line 328:
	Instead one assumes a correlation of <math>\rho=0</math> in the parent distribution and then compares the sample value of <math>R</math> with what you would get if there were no correlation.		Instead one assumes a correlation of <math>\rho=0</math> in the parent distribution and then compares the sample value of <math>R</math> with what you would get if there were no correlation.

−	The smaller <math>R</math> is the more likely that the data are correlated and that the linear fit is correct.	+	The smaller <math>R</math> is the more likely that the data are un-correlated and that the linear fit is correct.

← Older revision		Revision as of 19:19, 21 March 2014
Line 150:		Line 150:
	If the uncertainty in all the measurements is not the same then we need to insert <math>\sigma_i</math> back into the system of equations.		If the uncertainty in all the measurements is not the same then we need to insert <math>\sigma_i</math> back into the system of equations.

−	~~//%This section may have an error. Matrix elements of be in position 22 should be \sigma^2??~~
	: <math>A = \frac{\left\| \begin{array}{cc} \sum\frac{ y_i}{\sigma_i^2} & \sum\frac{ x_i}{\sigma_i^2}\\ \sum\frac{ x_i y_i}{\sigma_i^2} & \sum\frac{ x_i^2}{\sigma_i^2} \end{array}\right\|}{\left\| \begin{array}{cc} \sum \frac{1}{\sigma_i^2} & \sum \frac{x_i}{\sigma_i^2}\\ \sum \frac{x_i}{\sigma_i^2} & \sum \frac{x_i^2}{\sigma_i^2} \end{array}\right\|} \;\;\;\; B = \frac{\left\| \begin{array}{cc} \sum \frac{1}{\sigma_i^2} & \sum \frac{ y_i}{\sigma_i^2}\\ \sum \frac{x_i}{\sigma_i^2} & \sum \frac{x_i y_i}{\sigma_i^2} \end{array}\right\|}{\left\| \begin{array}{cc} \sum \frac{1}{\sigma_i^2} & \sum \frac{x_i}{\sigma_i^2}\\ \sum \frac{x_i}{\sigma_i^2} & \sum \frac{x_i^2}{\sigma_i^2} \end{array}\right\|}</math>		: <math>A = \frac{\left\| \begin{array}{cc} \sum\frac{ y_i}{\sigma_i^2} & \sum\frac{ x_i}{\sigma_i^2}\\ \sum\frac{ x_i y_i}{\sigma_i^2} & \sum\frac{ x_i^2}{\sigma_i^2} \end{array}\right\|}{\left\| \begin{array}{cc} \sum \frac{1}{\sigma_i^2} & \sum \frac{x_i}{\sigma_i^2}\\ \sum \frac{x_i}{\sigma_i^2} & \sum \frac{x_i^2}{\sigma_i^2} \end{array}\right\|} \;\;\;\; B = \frac{\left\| \begin{array}{cc} \sum \frac{1}{\sigma_i^2} & \sum \frac{ y_i}{\sigma_i^2}\\ \sum \frac{x_i}{\sigma_i^2} & \sum \frac{x_i y_i}{\sigma_i^2} \end{array}\right\|}{\left\| \begin{array}{cc} \sum \frac{1}{\sigma_i^2} & \sum \frac{x_i}{\sigma_i^2}\\ \sum \frac{x_i}{\sigma_i^2} & \sum \frac{x_i^2}{\sigma_i^2} \end{array}\right\|}</math>

@@ Line 150: / Line 150: @@
 If the uncertainty in all the measurements is not the same then we need to insert <math>\sigma_i</math> back into the system of equations.
-:%This section may have an error. Matrix elements of be in position 22 should be \sigma^2??
+//%This section may have an error. Matrix elements of be in position 22 should be \sigma^2??
 : <math>A = \frac{\left| \begin{array}{cc} \sum\frac{ y_i}{\sigma_i^2} & \sum\frac{ x_i}{\sigma_i^2}\\ \sum\frac{ x_i y_i}{\sigma_i^2} & \sum\frac{ x_i^2}{\sigma_i^2} \end{array}\right|}{\left| \begin{array}{cc} \sum \frac{1}{\sigma_i^2} & \sum \frac{x_i}{\sigma_i^2}\\ \sum \frac{x_i}{\sigma_i^2}  & \sum \frac{x_i^2}{\sigma_i^2} \end{array}\right|} \;\;\;\; B = \frac{\left| \begin{array}{cc} \sum \frac{1}{\sigma_i^2} & \sum \frac{ y_i}{\sigma_i^2}\\ \sum \frac{x_i}{\sigma_i^2}  & \sum \frac{x_i y_i}{\sigma_i^2} \end{array}\right|}{\left| \begin{array}{cc} \sum \frac{1}{\sigma_i^2} & \sum \frac{x_i}{\sigma_i^2}\\ \sum \frac{x_i}{\sigma_i^2}  & \sum \frac{x_i^2}{\sigma_i^2} \end{array}\right|}</math>