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GettingStarted.html
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.S12 { margin: 10px 10px 9px 4px; padding: 0px; line-height: 21px; min-height: 0px; white-space: pre-wrap; color: rgb(0, 0, 0); font-family: Helvetica, Arial, sans-serif; font-style: normal; font-size: 14px; font-weight: 400; text-align: left; }</style></head><body><div class = rtcContent><h1 class = 'S0'><span style=' font-weight: bold;'>Generalized chi-square distribution</span><span> · Getting started</span></h1><div class = 'S1'><span>The generalized chi-square variable is a quadratic form of a normal variable, or equivalently, a linear sum of independent non-central chi-square variables and a normal variable.</span></div><div class = 'S1'><span>Look into each function code or type </span><span style=' font-family: monospace;'>help functionname</span><span> for more features and documentation.</span></div><h2 class = 'S2'><span>Calculate mean and variance</span></h2><div class="CodeBlock"><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre;"><span style="color: rgb(2, 128, 9);">% gx2 parameters</span></span></div></div><div class="inlineWrapper"><div class = 'S4'><span style="white-space: pre;"><span>lambda=[1 -10 2];</span></span></div></div><div class="inlineWrapper"><div class = 'S4'><span style="white-space: pre;"><span>m=[1 2 3];</span></span></div></div><div class="inlineWrapper"><div class = 'S4'><span style="white-space: pre;"><span>delta=[2 3 7];</span></span></div></div><div class="inlineWrapper"><div class = 'S4'><span style="white-space: pre;"><span>sigma=5;</span></span></div></div><div class="inlineWrapper"><div class = 'S4'><span style="white-space: pre;"><span>c=10;</span></span></div></div><div class="inlineWrapper"><div class = 'S4'></div></div><div class="inlineWrapper outputs"><div class = 'S5'><span style="white-space: pre;"><span>[mu,v]=gx2stat(lambda,m,delta,sigma,c)</span></span></div><div class = 'S6'><div class='variableElement' style='font-family: Menlo, Monaco, Consolas, "Courier New", monospace; font-size: 12px; '>mu = -17</div><div class='variableElement' style='font-family: Menlo, Monaco, Consolas, "Courier New", monospace; font-size: 12px; '>v = 1771</div></div></div></div><h2 class = 'S2'><span>Generate random samples</span></h2><div class="CodeBlock"><div class="inlineWrapper"><div class = 'S7'><span style="white-space: pre;"><span>r=gx2rnd(lambda,m,delta,sigma,c,[1 1e4]);</span></span></div></div></div><h2 class = 'S2'><span>Calculate pdf and cdf</span></h2><div class="CodeBlock"><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre;"><span>x=[10 25];</span></span></div></div><div class="inlineWrapper outputs"><div class = 'S5'><span style="white-space: pre;"><span>f=gx2pdf(x,lambda,m,delta,sigma,c)</span></span></div><div class = 'S6'><div class="inlineElement eoOutputWrapper embeddedOutputsVariableMatrixElement" uid="EC27C71E" data-testid="output_2" data-width="567" style="width: 597px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="matrixElement veSpecifier" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="veVariableName variableNameElement double" style="width: 567px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="headerElementClickToInteract" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><span style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">f = </span><span class="veVariableValueSummary veMetaSummary" style="white-space: normal; font-style: normal; color: rgb(179, 179, 179); font-size: 12px;">1×2</span></div></div><div class="valueContainer" data-layout="{"columnWidth":66,"totalColumns":2,"totalRows":1,"charsPerColumn":10}" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="variableValue" style="width: 134px; white-space: pre; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"> 0.0121 0.0088
</div><div class="horizontalEllipsis hide" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"></div><div class="verticalEllipsis hide" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"></div></div></div></div></div></div><div class="inlineWrapper outputs"><div class = 'S8'><span style="white-space: pre;"><span class="warning_squiggle_rte956947607">p</span><span>=gx2cdf(x,lambda,m,delta,sigma,c,</span><span style="color: rgb(170, 4, 249);">'AbsTol'</span><span>,0,</span><span style="color: rgb(170, 4, 249);">'RelTol'</span><span>,1e-4)</span></span></div><div class = 'S6'><div class="inlineElement eoOutputWrapper embeddedOutputsVariableMatrixElement" uid="C3C8DE1E" data-testid="output_3" data-width="567" style="width: 597px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="matrixElement veSpecifier" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="veVariableName variableNameElement double" style="width: 567px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="headerElementClickToInteract" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><span style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">p = </span><span class="veVariableValueSummary veMetaSummary" style="white-space: normal; font-style: normal; color: rgb(179, 179, 179); font-size: 12px;">1×2</span></div></div><div class="valueContainer" data-layout="{"columnWidth":66,"totalColumns":2,"totalRows":1,"charsPerColumn":10}" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="variableValue" style="width: 134px; white-space: pre; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"> 0.7150 0.8790
</div><div class="horizontalEllipsis hide" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"></div><div class="verticalEllipsis hide" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"></div></div></div></div></div></div></div><h3 class = 'S9'><span>Plot pdf and sample histogram</span></h3><div class="CodeBlock"><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre;"><span>fplot(@(x) gx2pdf(x,lambda,m,delta,sigma,c))</span></span></div></div><div class="inlineWrapper"><div class = 'S4'><span style="white-space: pre;"><span>hold </span><span style="color: rgb(170, 4, 249);">on</span><span>; histogram(r,</span><span style="color: rgb(170, 4, 249);">'normalization'</span><span>,</span><span style="color: rgb(170, 4, 249);">'pdf'</span><span>,</span><span style="color: rgb(170, 4, 249);">'displaystyle'</span><span>,</span><span style="color: rgb(170, 4, 249);">'stairs'</span><span>)</span></span></div></div><div class="inlineWrapper"><div class = 'S4'><span style="white-space: pre;"><span>xline(mu,</span><span style="color: rgb(170, 4, 249);">'-'</span><span>,{</span><span style="color: rgb(170, 4, 249);">'\mu \pm \sigma'</span><span>},</span><span style="color: rgb(170, 4, 249);">'labelorientation'</span><span>,</span><span style="color: rgb(170, 4, 249);">'horizontal'</span><span>);</span></span></div></div><div class="inlineWrapper"><div class = 'S4'><span style="white-space: pre;"><span>xline(mu-sqrt(v),</span><span style="color: rgb(170, 4, 249);">'-'</span><span>); xline(mu+sqrt(v),</span><span style="color: rgb(170, 4, 249);">'-'</span><span>);</span></span></div></div><div class="inlineWrapper"><div class = 'S10'><span style="white-space: pre;"><span>xlim([mu-3*sqrt(v),mu+3*sqrt(v)]); ylim([0 .015]); ylabel </span><span style="color: rgb(170, 4, 249);">'pdf'</span></span></div></div></div><h3 class = 'S9'><span>Plot cdf</span></h3><div class="CodeBlock"><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre;"><span>figure; fplot(@(x) gx2cdf(x,lambda,m,delta,sigma,c));</span></span></div></div><div class="inlineWrapper"><div class = 'S4'><span style="white-space: pre;"><span>xline(mu,</span><span style="color: rgb(170, 4, 249);">'-'</span><span>,{</span><span style="color: rgb(170, 4, 249);">'\mu \pm \sigma'</span><span>},</span><span style="color: rgb(170, 4, 249);">'labelorientation'</span><span>,</span><span style="color: rgb(170, 4, 249);">'horizontal'</span><span>);</span></span></div></div><div class="inlineWrapper"><div class = 'S4'><span style="white-space: pre;"><span>xline(mu-sqrt(v),</span><span style="color: rgb(170, 4, 249);">'-'</span><span>); xline(mu+sqrt(v),</span><span style="color: rgb(170, 4, 249);">'-'</span><span>);</span></span></div></div><div class="inlineWrapper"><div class = 'S10'><span style="white-space: pre;"><span>xlim([mu-3*sqrt(v),mu+3*sqrt(v)]); ylim([0 1]); xlabel </span><span style="color: rgb(170, 4, 249);">x</span><span>; ylabel </span><span style="color: rgb(170, 4, 249);">'cdf'</span></span></div></div></div><h2 class = 'S2'><span>Calculate inverse cdf</span></h2><div class="CodeBlock"><div class="inlineWrapper outputs"><div class = 'S11'><span style="white-space: pre;"><span>x=gx2inv([0.5 0.9],lambda,m,delta,sigma,c)</span></span></div><div class = 'S6'><div class="inlineElement eoOutputWrapper embeddedOutputsVariableMatrixElement" uid="225BF16C" data-testid="output_4" data-width="567" style="width: 597px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="matrixElement veSpecifier" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="veVariableName variableNameElement double" style="width: 567px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="headerElementClickToInteract" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><span style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">x = </span><span class="veVariableValueSummary veMetaSummary" style="white-space: normal; font-style: normal; color: rgb(179, 179, 179); font-size: 12px;">1×2</span></div></div><div class="valueContainer" data-layout="{"columnWidth":66,"totalColumns":2,"totalRows":1,"charsPerColumn":10}" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="variableValue" style="width: 134px; white-space: pre; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"> -8.7657 27.5320
</div><div class="horizontalEllipsis hide" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"></div><div class="verticalEllipsis hide" style="white-space: nowrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"></div></div></div></div></div></div></div><h2 class = 'S2'><span>Distribution of quadratic form of a normal variable</span></h2><div class="CodeBlock"><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre;"><span style="color: rgb(2, 128, 9);">% normal parameters</span></span></div></div><div class="inlineWrapper"><div class = 'S4'><span style="white-space: pre;"><span>mu=[1;2]; </span><span style="color: rgb(2, 128, 9);">% mean</span></span></div></div><div class="inlineWrapper"><div class = 'S4'><span style="white-space: pre;"><span>v=[2 1; 1 3]; </span><span style="color: rgb(2, 128, 9);">% covariance matrix</span></span></div></div><div class="inlineWrapper"><div class = 'S10'></div></div></div><div class = 'S12'><span texencoding="q(\mathbf{x})=(x_1+x_2)^2-x_1-1" style="vertical-align:-6px"><img src="data:image/png;base64,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" width="158.5" height="21" alt="q(\mathbf{x})=(x_1+x_2)^2-x_1-1" /></span><span style=' font-family: monospace;'> = [x1;x2]'*[1 1; 1 1]*[x1;x2] + [-1;0]'*[x1;x2] -1</span></div><div class="CodeBlock"><div class="inlineWrapper"><div class = 'S3'><span style="white-space: pre;"><span>quad.q2=[1 1; 1 1];</span></span></div></div><div class="inlineWrapper"><div class = 'S4'><span style="white-space: pre;"><span>quad.q1=[-1;0];</span></span></div></div><div class="inlineWrapper"><div class = 'S4'><span style="white-space: pre;"><span>quad.q0=-1;</span></span></div></div><div class="inlineWrapper"><div class = 'S4'></div></div><div class="inlineWrapper"><div class = 'S4'><span style="white-space: pre;"><span style="color: rgb(2, 128, 9);">% get gx2 parameters corr. to this quadratic form</span></span></div></div><div class="inlineWrapper outputs"><div class = 'S5'><span style="white-space: pre;"><span>[lambda,m,delta,sigma,c]=gx2_params_norm_quad(mu,v,quad)</span></span></div><div class = 'S6'><div class='variableElement' style='font-family: Menlo, Monaco, Consolas, "Courier New", monospace; font-size: 12px; '>lambda = 7.0000</div><div class='variableElement' style='font-family: Menlo, Monaco, Consolas, "Courier New", monospace; font-size: 12px; '>m = 1</div><div class='variableElement' style='font-family: Menlo, Monaco, Consolas, "Courier New", monospace; font-size: 12px; '>delta = 1.1086</div><div class='variableElement' style='font-family: Menlo, Monaco, Consolas, "Courier New", monospace; font-size: 12px; '>sigma = 0.8452</div><div class='variableElement' style='font-family: Menlo, Monaco, Consolas, "Courier New", monospace; font-size: 12px; '>c = -0.7602</div></div></div></div><div class = 'S12'><span texencoding="p(q(\mathbf{x})<3)" style="vertical-align:-5px"><img src="data:image/png;base64,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" width="76.5" height="19" alt="p(q(x)<3)" /></span></div><div class="CodeBlock"><div class="inlineWrapper outputs"><div class = 'S11'><span style="white-space: pre;"><span>p=gx2cdf(3,lambda,m,delta,sigma,c)</span></span></div><div class = 'S6'><div class='variableElement' style='font-family: Menlo, Monaco, Consolas, "Courier New", monospace; font-size: 12px; '>p = 0.3351</div></div></div></div></div>
<br>
<!--
##### SOURCE BEGIN #####
%% *Generalized chi-square distribution* · Getting started
% The generalized chi-square variable is a quadratic form of a normal variable,
% or equivalently, a linear sum of independent non-central chi-square variables
% and a normal variable.
%
% Look into each function code or type |help functionname| for more features
% and documentation.
%% Calculate mean and variance
% gx2 parameters
lambda=[1 -10 2];
m=[1 2 3];
delta=[2 3 7];
sigma=5;
c=10;
[mu,v]=gx2stat(lambda,m,delta,sigma,c)
%% Generate random samples
r=gx2rnd(lambda,m,delta,sigma,c,[1 1e4]);
%% Calculate pdf and cdf
x=[10 25];
f=gx2pdf(x,lambda,m,delta,sigma,c)
p=gx2cdf(x,lambda,m,delta,sigma,c,'AbsTol',0,'RelTol',1e-4)
% Plot pdf and sample histogram
fplot(@(x) gx2pdf(x,lambda,m,delta,sigma,c))
hold on; histogram(r,'normalization','pdf','displaystyle','stairs')
xline(mu,'-',{'\mu \pm \sigma'},'labelorientation','horizontal');
xline(mu-sqrt(v),'-'); xline(mu+sqrt(v),'-');
xlim([mu-3*sqrt(v),mu+3*sqrt(v)]); ylim([0 .015]); ylabel 'pdf'
% Plot cdf
figure; fplot(@(x) gx2cdf(x,lambda,m,delta,sigma,c));
xline(mu,'-',{'\mu \pm \sigma'},'labelorientation','horizontal');
xline(mu-sqrt(v),'-'); xline(mu+sqrt(v),'-');
xlim([mu-3*sqrt(v),mu+3*sqrt(v)]); ylim([0 1]); xlabel x; ylabel 'cdf'
%% Calculate inverse cdf
x=gx2inv([0.5 0.9],lambda,m,delta,sigma,c)
%% Distribution of quadratic form of a normal variable
% normal parameters
mu=[1;2]; % mean
v=[2 1; 1 3]; % covariance matrix
%%
% $q(\mathbf{x})=(x_1+x_2)^2-x_1-1$ |= [x1;x2]'*[1 1; 1 1]*[x1;x2] + [-1;0]'*[x1;x2]
% -1|
quad.q2=[1 1; 1 1];
quad.q1=[-1;0];
quad.q0=-1;
% get gx2 parameters corr. to this quadratic form
[lambda,m,delta,sigma,c]=gx2_params_norm_quad(mu,v,quad)
%%
% $$p(q(\mathbf{x})<3)$$
p=gx2cdf(3,lambda,m,delta,sigma,c)
##### SOURCE END #####
--></body></html>