I'm studying my Biomechanics course and I need to reproduce results given in the book "Biomechanics and motor control in human movement by David A. Winter" Chapter 7, pages 186-187 numerically using Octave or Matlab.
I'm unable to solve the matrix equation (please see screenshots):
From what I know about matrices is that they are orthogonal (resulting from Kardan rotation sequence x-y'-z'').
In the aforementioned matrices, terms named s1, s2, s3 are the sines of theta1, theta2, and theta3, respectively and c1, c2, and c3 are the cosines of theta1, theta2, and theta3, respectively.
Below is the code I've tried in the Octave-online using symbolic toolbox and solve function.
GA = [0.5974 -0.7873 0.1515;
0.800 0.5969 -0.00550;
-0.0472 0.1544 0.9868]
% syms t1 t2 t3 real
% GAS = [(cosd(t2)*cosd(t3)) (sind(t3)*cosd(t1)+sind(t1)*sind(t2)*cosd(t3)) (sind(t1)*sind(t3)-cosd(t1)*sind(t2)*cosd(t3)); (-cosd(t2)*sind(t3)) (cosd(t1)*cosd(t3)-sind(t1)*sind(t2)*sind(t3)) (sind(t1)*cosd(t3)+cosd(t1)*sind(t2)*sind(t3)); (sind(t2)) (-sind(t1)*cosd(t2)) (cosd(t1)*cosd(t2))]
% [t1, t2, t3] = solve(GAS == GA, t1, t2, t3)
syms c1 c2 c3 s1 s2 s3 real
GAS = [c2*c3 s3*c1+s1*s2*c3 s1*s3-c1*s2*c3; -c2*s3 c1*c3-s1*s2*s3 s1*c3+c1*s2*s3; s2 -s1*c2 c1*c2]
S = solve(GAS == GA, c1, c2, c3, s1, s2, s3)
The result from Octave-Online (using both versions of code results in an empty structure).
GA =
0.5974000 -0.7873000 0.1515000
0.8000000 0.5969000 -0.0055000
-0.0472000 0.1544000 0.9868000
Symbolic pkg v2.8.0: Python communication link active, SymPy v1.5.
GAS = (sym 3×3 matrix)
⎡c₂⋅c₃ c₁⋅s₃ + c₃⋅s₁⋅s₂ -c₁⋅c₃⋅s₂ + s₁⋅s₃⎤
⎢ ⎥
⎢-c₂⋅s₃ c₁⋅c₃ - s₁⋅s₂⋅s₃ c₁⋅s₂⋅s₃ + c₃⋅s₁ ⎥
⎢ ⎥
⎣ s₂ -c₂⋅s₁ c₁⋅c₂ ⎦
warning: passing floating-point values to sym is dangerous, see "help sym"
warning: called from
double_to_sym_heuristic at line 50 column 7
sym at line 379 column 13
numeric_array_to_sym at line 36 column 16
sym at line 359 column 7
eq at line 91 column 5
warning: passing floating-point values to sym is dangerous, see "help sym"
warning: called from
double_to_sym_heuristic at line 50 column 7
sym at line 379 column 13
numeric_array_to_sym at line 36 column 16
sym at line 359 column 7
eq at line 91 column 5
warning: passing floating-point values to sym is dangerous, see "help sym"
warning: called from
double_to_sym_heuristic at line 50 column 7
sym at line 379 column 13
numeric_array_to_sym at line 36 column 16
sym at line 359 column 7
eq at line 91 column 5
warning: passing floating-point values to sym is dangerous, see "help sym"
warning: called from
double_to_sym_heuristic at line 50 column 7
sym at line 379 column 13
numeric_array_to_sym at line 36 column 16
sym at line 359 column 7
eq at line 91 column 5
warning: passing floating-point values to sym is dangerous, see "help sym"
warning: called from
double_to_sym_heuristic at line 50 column 7
sym at line 379 column 13
numeric_array_to_sym at line 36 column 16
sym at line 359 column 7
eq at line 91 column 5
warning: passing floating-point values to sym is dangerous, see "help sym"
warning: called from
double_to_sym_heuristic at line 50 column 7
sym at line 379 column 13
numeric_array_to_sym at line 36 column 16
sym at line 359 column 7
eq at line 91 column 5
warning: passing floating-point values to sym is dangerous, see "help sym"
warning: called from
double_to_sym_heuristic at line 50 column 7
sym at line 379 column 13
numeric_array_to_sym at line 36 column 16
sym at line 359 column 7
eq at line 91 column 5
warning: passing floating-point values to sym is dangerous, see "help sym"
warning: called from
double_to_sym_heuristic at line 50 column 7
sym at line 379 column 13
numeric_array_to_sym at line 36 column 16
sym at line 359 column 7
eq at line 91 column 5
warning: passing floating-point values to sym is dangerous, see "help sym"
warning: called from
double_to_sym_heuristic at line 50 column 7
sym at line 379 column 13
numeric_array_to_sym at line 36 column 16
sym at line 359 column 7
eq at line 91 column 5
S = {}(0x0)
Cheers,
SirDickens
Related
I have a big matrix (8656x25960) with some speckle noise within it. I used the findpeaks tool in order to find in what columns I indeed have peaks above a certain threshold. The output of the findspeaks tool is a matrix containing all of the bad columns, for example -
loc =
Columns 1 through 6
30 51 155 307 333 338
Columns 7 through 12
642 955 1409 1567 1728 1730
Columns 13 through 18
2332 2546 2615 2685 2806 2995
Columns 19 through 24
3002 3122 3124 3164 3690 4176
Columns 25 through 30
4430 4475 4539 5142 5155 5244
Columns 31 through 36
5246 5941 5943 6114 6486 6922
Columns 37 through 42
7165 7169 7460 7587 7647 8944
Columns 43 through 44
12754 13693
How can I use those columns numbers with the original matrix and replace the values of this 'bad' column with the value 0 (for example).
Hoping I'm clear enough.
For row vector Ioc simply use indexing:
yourmatrix(:,Ioc) = 0;
In MATLAB (R2015a), I have two large matrices that can be simplified as:
A = [ 24 10 830; 24 15 830; 150 17 945; 231 40 1130; 231 45 1130]
(note that in A, column 3 is the time for the event cataloged in column 1) and
B = [24 13; 150 29; 231 43]
How can I match and copy the time-data from column 3 in matrix A, to a matching and filtered event column in matrix B?
For example, I want the value 24 from first column in B to be matched with value 24 from first column in A, and then copy the corresponding time-data in A's third column (for 24 it's 830, for 150 it's 945 etc.) into B. This should result into our new B with the time-data from A:
B = [24 13 830; 150 29 945; 231 43 1130]
I relatively new to MATLAB so any help is much appreciated!
First find the location of the elements in the first row of B in the first row of A using the ismember function. And then use those locations to construct the new matrix.
[~,Locb] = ismember(B(:,1),A(:,1));
Bnew = [B A(Locb,3)]
Bnew =
24 13 830
150 29 945
231 43 1130
This is a fast way that comes to my mind. There might be some singularities that needed to be checked more thoroughly.
I have data of integers in x = 500 X 612 matrix. I need a new variable xx in a 500 X 612 matrix but I need to apply cumsum along each row (500) across 12 column steps and applying cumsum like this 51 times --> 500 X (12 X 51) matrix. Then I need a for loop to produce 51 plots of the 500 rows and 12 columns of the cumsum time series. thank you!
I will rephrase what the question is asking to benefit those who are reading.
The OP wishes to segment a matrix into chunks by splitting up the matrix into a bunch of columns. A cumsum is applied to each row individually for each column and are then concatenated together to build a final matrix. As such, given this source matrix:
x =
1 2 3 4 5 6 7 8 9 10 11 12
13 14 15 16 17 18 19 20 21 22 23 24
Supposing that we wish to split up the matrix by columns 3, 6 and 9 and 12, we will have four chunks to work with. We do a cumsum on each of these blocks individually and piece the final result together. So the result would like the following:
xx =
1 3 6 4 9 15 7 15 24 10 21 33
13 27 42 16 33 51 19 39 60 22 45 69
First, you need to determine how many columns you want to break up the matrix into. In your case, we wish to segment the matrix into 4 chunks: Columns 1 - 3, columns 4 - 6, columns 7 - 9, and columns 10 - 12. As such, I'm going to reshape this matrix so that each column is an individual row from a chunk in this matrix. We then apply cumsum over this reshaped matrix and we then reshape it back to what you had originally.
Therefore, do this:
num_chunks = 4; %// Columns 3, 6, 9, 12
divide_point = size(x,2) / num_chunks; %// Determine how many elements are in a row for a cumsum
x_reshape = reshape(x.', divide_point, []); %// Get reshaped matrix
xy = cumsum(x_reshape); %// cumsum over all columns individually
xx = reshape(xy, size(x,2), size(x,1)).'; %// Reconstruct matrix
In the third line of code, x_reshape = reshape(x.', divide_point, []); may seem a bit daunting, but it's actually not that bad. I had to transpose the matrix first because you want to take each row of a chunk and place them into individual columns so we can perform a cumsum on each column. When you reshape something in MATLAB, it collects values column-wise and reshapes the input into an output of a specified size. Therefore, to collect the rows, we need to collect row-wise and so we must transpose this matrix. Next, divide_point tells you how many elements we have for a single row in one chunk. As such, we want to construct a matrix that is of size divide_point x N where divide_point tells you how many elements we have in a row of a chunk and N is the total number of rows over all chunks. Because I don't want to calculate how many there are (am rather lazy actually....), the [] syntax is to automatically infer this number so that we can get a reshaped matrix that respects the total number of elements in the original input. We then perform cumsum on each of these columns, and then we need to reshape this back into the original shape of the input. With this, we use reshape again on the cumsum result, but in order to get it back into the row-order that you want, we have to determine the transpose as reshape takes values in column-major order, then re-transpose that result.
We get:
xx =
1 3 6 4 9 15 7 15 24 10 21 33
13 27 42 16 33 51 19 39 60 22 45 69
In general, the total number of elements to sum over for a row needs to be evenly divisible by the total number of columns that your matrix contains. For example, given the above, if you were to try to segment this matrix into 5 chunks, you would certainly get an error as the number of rows to cumsum over is not symmetric.
As another example, let's say we wanted to break up the matrix into 6 chunks. Therefore, by setting num_chunks = 6, we get:
xx =
1 3 3 7 5 11 7 15 9 19 11 23
13 27 15 31 17 35 19 39 21 43 23 47
You can see that cumsum restarts at every second column, as we desired 6 chunks and to get 6 chunks with a matrix of 12 columns, a chunk is created at every second column.
I am a beginner with MATLAB and I am struggling with this assignment. Can anyone guide me through it?
Consider the data given below:
x = [ 1 , 48 , 81 , 2 , 10 , 25 , ,14 , 18 , 53 , 41, 56, 89,0, 1000, , ...
34, 47, 455, 21, , 22, 100 ];
Once the data is loaded, see if you can find any:
Outliers or
Missing data in the data file
Correct the missing values using median, mode and noisy data using median binning, mean binning and bin boundaries.
This isn't so bad. First off, take a look at the distribution of your data. You can see that the majority of your data has double digits. The outliers are those with single digits, or those that are way larger than double digits. Mind you, this is totally subjective so someone else may tell you that the single digits are part of your data too. Also, the missing data are those numbers that are spaces in between the commas. Let's write some MATLAB code and change these to NaN (or not-a-number), because if you try copying and pasting this code directly into MATLAB, it will give you a syntax error because if you are explicitly defining numbers this way, you have to be sure all of them are there.
To do this, use regexprep so that any parts of this string that have a comma, space, then another comma, put a NaN in between. To do this, we need to put this statement as a string first. We then use eval to convert this string to an actual MATLAB statement:
x = '[ 1 , 48 , 81 , 2 , 10 , 25 , ,14 , 18 , 53 , 41, 56, 89,0, 1000, , 34, 47, 455, 21, , 22, 100 ];'
y = eval(regexprep(x, ', ,', ', NaN, '));
If we display this data, we get:
y =
Columns 1 through 6
1 48 81 2 10 25
Columns 7 through 12
NaN 14 18 53 41 56
Columns 13 through 18
89 0 1000 NaN 34 47
Columns 19 through 23
455 21 NaN 22 100
As such, to answer our first question, any values that are missing are denoted as NaN and those numbers that are bigger than double digits are outliers.
For the next question, we simply extract those values that are not missing, calculate the mean and median of what is not missing, and fill in those NaN values with the mean and median. For the bin boundaries, this is the same thing as using the values to the left (or right... depends on your definition, but let's use left) of the missing value and fill those in. As such:
yMissing = isnan(y); %// Which values are missing?
y_noNaN = y(~yMissing); %// Extract the non-missing values
meanY = mean(y_noNaN); %// Get the mean
medianY = median(y_noNaN); %// Get the median
%// Output - Fill in missing values with median
yMedian = y;
yMedian(yMissing) = medianY;
%// Same for mean
yMean = y;
yMean(yMissing) = meanY;
%// Bin boundaries
yBinBound = y;
yBinBound(yMissing) = y(find(yMissing)-1);
The mean and median for the data of the non-missing values is:
meanY =
105.8500
medianY =
37.5000
The outputs for each of these, in addition to the original data with the missing values looks like:
format bank; %// Do this to show just the first two decimal places for compact output
format compact;
y =
Columns 1 through 5
1 48 81 2 10
Columns 6 through 10
25 NaN 14 18 53
Columns 11 through 15
41 56 89 0 1000
Columns 16 through 20
NaN 34 47 455 21
Columns 21 through 23
NaN 22 100
yMean =
Columns 1 through 5
1.00 48.00 81.00 2.00 10.00
Columns 6 through 10
25.00 105.85 14.00 18.00 53.00
Columns 11 through 15
41.00 56.00 89.00 0 1000.00
Columns 16 through 20
105.85 34.00 47.00 455.00 21.00
Columns 21 through 23
105.85 22.00 100.00
yMedian =
Columns 1 through 5
1.00 48.00 81.00 2.00 10.00
Columns 6 through 10
25.00 37.50 14.00 18.00 53.00
Columns 11 through 15
41.00 56.00 89.00 0 1000.00
Columns 16 through 20
37.50 34.00 47.00 455.00 21.00
Columns 21 through 23
37.50 22.00 100.00
yBinBound =
Columns 1 through 5
1.00 48.00 81.00 2.00 10.00
Columns 6 through 10
25.00 25.00 14.00 18.00 53.00
Columns 11 through 15
41.00 56.00 89.00 0 1000.00
Columns 16 through 20
1000.00 34.00 47.00 455.00 21.00
Columns 21 through 23
21.00 22.00 100.00
If you take a look at each of the output values, this fills in our data with the mean, median and also the bin boundaries as per the question.
This question is an outgrowth of MatLab (or any other language) to convert a matrix or a csv to put 2nd column values to the same row if 1st column value is the same?
If
A = [2 3 234 ; 2 44 33; 2 12 22; 3 123 99; 3 1232 45; 5 224 57]
1st column | 2nd column | 3rd column
2 3 234
2 44 33
2 12 22
3 123 99
3 1232 45
5 224 57
then running
[U ix iu] = unique(A(:,1) );
r= accumarray( iu, A(:,2:3), [], #(x) {x'} )
will show me the error
Error using accumarray
Second input VAL must be a vector with one element for each row in SUBS, or a
scalar.
I want to make
1st col | 2nd col | 3rd col | 4th col | 5th col | 6th col| 7th col
2 3 234 44 33 12 22
3 123 99 1232 45
5 224 57
I know how to do it using for and if, but that spends too much time for big data.
How can I do this?
Thank you in advance!
You're misusing accumarray in the solution provided to your previous question. The first parameter iu is the vector of indices and the second parameter should be a vector of values, of the same length. What you did here is specify a matrix as the second parameter, which in fact has twice more values than indices in iu.
What you need to do in order to make it work is create a vector of indices both for the second column and for the third column (they are the same indices, not coincidentally!) and specify a matching column vector of values, like so:
[U, ix, iu] = unique(A(:,1));
vals = reshape(A(:, 2:end).', [], 1); %'// Columnize values
subs = reshape(iu(:, ones(size(A, 2) - 1, 1)).', [], 1); %'// Replicate indices
r = accumarray(subs, vals, [], #(x){x'});
This solution is generalized for any number of columns that you want to pass to accumarray.