Added documentation and demos

macros/arch_fit.md

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+#  arch_fit
+## Calling Sequence
+
+- ` [a, b] = arch_fit (y, x, p) `
+- ` [a, b] = arch_fit (y, x, p, iter, gamma, a0, b0) `
+
+## Description
+Fit an ARCH regression model to the time series y using the scoring algorithm in Engle’s original ARCH paper.
+
+The model is
+
+y(t) = b(1) * x(t,1) + … + b(k) * x(t,k) + e(t),
+h(t) = a(1) + a(2) * e(t-1)^2 + … + a(p+1) * e(t-p)^2
+in which e(t) is N(0, h(t)), given a time-series vector y up to time t-1 and a matrix of (ordinary) regressors x up to t. The order of the regression of the residual variance is specified by p.
+
+If invoked as arch_fit (y, k, p) with a positive integer k, fit an ARCH(k, p) process, i.e., do the above with the t-th row of x given by
+
+[1, y(t-1), …, y(t-k)]
+Optionally, one can specify the number of iterations iter, the updating factor gamma, and initial values a0 and b0 for the scoring algorithm.
+##  Parameters
+- `y`(vector) :  A time-series data vector up to time t-1 .
+- `x` (Matrix):  A matrix of (ordinary) regressors x up to t.
+- `p` (scalar):  The order of the regression of the residual variance.
+- `iter` (scaler) : Number of iterations
+- `gamma` (real number) : updating factor
+- `a0 b0` (real numbers) : Initial values for the scoring algorithm
+## Dependencies: 
+ols autoreg_matrix
+
+
+## Examples
+1. 
+```scilab
+t = linspace(0,10,1000);
+
+y = cos(t);
+
+x = sin(t);
+
+[a,b] = arch_fit(y,x',1)
+
+```
+```output
+ pval  = 
+
+   0.0175825
+   0.9357096
+ lm  = 
+
+   0.0242719
+
+  ```
+
+2.
+```scilab
+t = linspace(0,10,1000);
+y = sin(t);
+x = [ zeros(1,300) ones(1,400) zeros(1,300);zeros(1,200) ones(1,200) zeros(1,600);zeros(1,100) ones(1,400) zeros(1,500)];
+[a,b] = arch_fit(y,x',3)
+```
+```output
+ a  = 
+
+   0.0020139
+   1.0353154
+  -0.0009762
+  -0.0334181
+ b  = 
+
+  -0.7849689
+   0.2267450
+   0.3037736
+```
+3.
+```scilab
+ t = linspace(-10,10,2000);
+y = exp(t);
+x = [sin(t);cos(t);tan(t)];
+[a,b] = arch_fit(y,x',5)
+
+```
+```output
+a  = 
+
+   69311.100
+   0.8883086
+   0.3352075
+   0.1473639
+  -0.0385322
+  -0.3485546
+ b  = 
+
+   334.06505
+  -1464.5987
+  -0.5365645
+```
+4.
+```scilab
+ t = linspace(-10,10,2000);
+y = exp(t);
+x = [sin(t);cos(t);tan(t)];
+ [a,b] = arch_fit(y,x',5,1000,0.32,[1;0;1;0;1;0],[1;0;1]);
+ a,b
+```
+```output
+ a  = 
+
+   0.0001950
+   1.1362016
+  -0.1143480
+   0.0180131
+  -0.0284946
+   0.0200169
+ b  = 
+
+   1.
+  -6.465D-22
+   1.
+
+```
+5.
+```scilab
+
+t= linspace(0,100,100);
+
+y = t .* t + 2*t + 3;
+
+ x = 3;
+
+[a,b] = arch_fit(y,x,1,1000,0.2,[1 ;1],[0; 0 ;0 ;0])
+```
+```output
+//Octave will return Nan vectors which means no solution  
+
+inv: Problem is singular. // implies no solution
+```

macros/arch_test.md

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+# arch_test
+## Description
+Perform a Lagrange Multiplier (LM) test for conditional heteroscedasticity.
+
+For a linear regression model
+
+y = x * b + e
+perform a Lagrange Multiplier (LM) test of the null hypothesis of no conditional heteroscedascity against the alternative of CH(p).
+
+I.e., the model is
+
+y(t) = b(1) * x(t,1) + … + b(k) * x(t,k) + e(t),
+given y up to t-1 and x up to t, e(t) is N(0, h(t)) with
+
+h(t) = v + a(1) * e(t-1)^2 + … + a(p) * e(t-p)^2,
+and the null is a(1) == … == a(p) == 0.
+
+If the second argument is a scalar integer, k, perform the same test in a linear autoregression model of order k, i.e., with
+
+[1, y(t-1), …, y(t-k)]
+as the t-th row of x.
+
+Under the null, LM approximately has a chisquare distribution with p degrees of freedom and pval is the p-value (1 minus the CDF of this distribution at LM) of the test.
+
+If no output argument is given, the p-value is displayed.
+## Calling Sequence
+
+- ` [pval, lm] = arch_test (y, x, p) `
+
+
+## Parameters
+- `y`: Array-like. Dependent variable of the regression model.
+- `x`: Array-like. Independent variables of the regression model.
+         If x is a scalar integer k, it represents the order of autoregression.
+- `p` : Integer. Number of lagged squared residuals to include in the heteroscedasticity model.
+
+
+Returns:
+- `pval`: Float. p-value of the LM test.
+- `lm`: Float. Lagrange Multiplier test statistic.
+
+
+## Dependencies: 
+ols , autoreg_matrix
+
+## Examples
+1. 
+```scilab
+t = linspace(0,10,1000);
+
+y = cos(t);
+
+x = sin(t);
+
+[pval,lm] = arch_test(y,x',1)
+
+```
+```output
+pval = 0
+lm = 229.01679
+```
+
+2.
+```scilab
+t = linspace(0,10,1000);
+y = sin(t);
+x = [ zeros(1,300) ones(1,400) zeros(1,300);zeros(1,200) ones(1,200) zeros(1,600);zeros(1,100) ones(1,400) zeros(1,500)];
+[pval,lm] = arch_test(y,x',3)
+```
+```output
+pval  = 
+
+   0.
+ lm  = 
+
+   447.99516
+
+```
+3.
+```scilab
+t = linspace(-10,10,2000);
+y = exp(t);
+x = [sin(t);cos(t);tan(t)];
+[pval,lm] = arch_test(y,x',5)
+```
+```output
+Warning :
+matrix is close to singular or badly scaled. rcond = 6.6096E-17
+ pval  = 
+
+   0.
+ lm  = 
+
+   16935.914
+
+```
+4.
+```scilab
+t= linspace(0,100,100);
+
+y = t .* t + 2*t + 3;
+
+ x = 3;
+
+[pval,lm] = arch_test(y,x,1)
+
+```
+```output
+ pval  = 
+
+   0.
+ lm  = 
+
+   208.60268
+
+```
+5.
+```scilab
+t= linspace(0,10,1000);
+x=[sin(t);cos(t);tan(t)];
+[pval,lm] = arch_test(t,x',1)
+```
+```output
+pval  = 
+
+   0.
+ lm  = 
+
+   345.23194
+
+```

macros/cohere.md

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+# cohere
+## Description
+Estimate (mean square) coherence of signals "x" and "y".
+Use the Welch (1967) periodogram/FFT method.
+Compatible with Matlab R11 cohere and earlier.
+See "help pwelch" for description of arguments, hints and references — especially hint (7) for Matlab R11 defaults.
+
+## Calling Sequence
+- ` [Pxx, freq] = cohere(x,y,Nfft,Fs,window,overlap,range,plot_type,detrend) `
+
+## Dependencies: 
+pwelch
+## Examples
+1. 
+```scilab
+subplot(2,2,1)
+t = linspace(0,10,1000); x = sin(t) ; y = cos(t);
+cohere(x,y)
+subplot(2,2,2)
+cohere(t,x,400)
+subplot(2,2,3)
+cohere(t,y,300,500)
+subplot(2,2,4)
+t = linspace(1,10,1000); x =sin(t);
+y = filter(0.23,x,t);
+cohere(x,y,500,100,6)
+```
+<img src="cohere_t_case1.svg">
+
+2.
+```scilab
+subplot(2,2,1)
+t = linspace(1,10,1000); x =cos(t);
+y = filter(0.9999,x,t);
+cohere(x,y,800,300,7,0.56)
+subplot(2,2,2)
+t = linspace(1,10,1000); x =filter(0.3245,cos(t),t); y = filter(0.0034,x,sin(t));
+cohere(x,y,700,1000,4,0.67,"half")
+```
+<img src="cohere_t_case2.svg">