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# Lensless light-field imaging through diffuser encoding

## Abstract

Microlens array-based light-field imaging has been one of the most commonly used and effective technologies to record high-dimensional optical signals for developing various potential high-performance applications in many fields. However, the use of a microlens array generally suffers from an intrinsic trade-off between the spatial and angular resolutions. In this paper, we concentrate on exploiting a diffuser to explore a novel modality for light-field imaging. We demonstrate that the diffuser can efficiently angularly couple incident light rays into a detected image without needing any lens. To characterize and analyse this phenomenon, we establish a diffuser-encoding light-field transmission model, in which four-dimensional light fields are mapped into two-dimensional images via a transmission matrix describing the light propagation through the diffuser. Correspondingly, a calibration strategy is designed to flexibly determine the transmission matrix, so that light rays can be computationally decoupled from a detected image with adjustable spatio-angular resolutions, which are unshackled from the resolution limitation of the sensor. The proof-of-concept approach indicates the possibility of using scattering media for lensless four-dimensional light-field recording and processing, not just for two- or three-dimensional imaging.

## Introduction

Conventional photography forms a two-dimensional image on a sensor, leading to loss of angular information. In contrast, light-field imaging can detect both spatial and angular information1,2. The angular information offers peculiar capabilities over conventional imaging, such as viewpoint shifting, post-capture refocusing, depth sensing, and depth-of-field extension. Consequently, light-field imaging has a growing range of applications involving light-field microscopy3,4,5,6,7, synthetic aperture imaging8,9, visual odometry and localization10,11, and many others.

For light-field imaging, a primary and critical step is how to effectively record angular-resolved light fields. According to the number of image sensors and exposures used, light-field imaging can be mainly classified into three categories. One is using multiple image sensors with a single exposure to simultaneously capture light-field samples from different viewpoints5,8,12. Such a system usually consists of a large number of cameras, which are bulky and expensive and thus not suitable for practical applications. Alternatively, light-field samples can be obtained using a single sensor and multiple exposures1,2,13. This approach can record light fields with high spatial and angular resolutions, but it is time-consuming and thus unsuitable for dynamic scenes.

The third approach uses a single image sensor with a single exposure to encode four-dimensional spatio-angular information into a two-dimensional detected image. More than a century ago, the concept of plenoptic cameras by adding a pinhole array or microlens array was proposed14,15. Currently, microlens array-based plenoptic cameras are commonly used for light-field imaging16,17,18, such as the commercially available products, Lytro and Raytrix. These devices can record unambiguous light fields in which a pixel corresponds to a single light ray and can conceptually be thought of as classical light-field imaging. However, classical light-field imaging involves a trade-off between the spatial and angular resolutions; the spatial resolution is in general tens to hundreds times smaller than the number of pixels used. In recent years, alternative techniques have been developed to encode a light ray on multiple pixels through modulation masks (e.g., attenuation masks19,20,21 and diffuser plates22), avoiding the resolution trade-off. A light field is no longer recorded directly but computationally retrieved from a detected image, which can be referred to as computational light-field imaging.

In this paper, we propose a novel modality for computational light-field imaging by using a diffuser as an encoder, without needing any lens. Through the diffuser, each sub-beam directionally emitted by a point source in the detectable field-of-view forms a distinguishable sub-image that covers a specific region on the sensor. These sub-images are combined into a unique pseudorandom pattern corresponding to the response of the system to the point source. Consequently, the system has the capability of encoding a light-field incident onto the diffuser. We establish a diffuser-encoding light-field transmission model to characterize the mapping of four-dimensional light fields to two-dimensional images, where a pixel collects and integrates contributions from different sub-beams. With the aid of the optical properties of the diffuser encoding, the light-field transmission matrix can be flexibly calibrated through a point source generated pattern. As a result, light fields are computationally reconstructed with adjustable spatio-angular resolutions, avoiding the resolution limitation of the sensor. Being significantly different from the existing approaches using diffusers for lensless two- or three-dimensional imaging23,24,25,26, our imaging modality provides a proof-of-concept approach in which scattering media can be exploited for recording high-dimensional optical signals, modewith which the intrinsic mechanism of light propagation can be further explored.

## Results

### Diffuser encoding

The diffuser used in the system (see Fig. 1) is a thin transparent phase plate with a statistically varying surface distribution. Through the diffuser, a temporally incoherent point source in the detectable field-of-view generates a high-contrast pseudorandom pattern on the sensor, as shown by the green and orange wireframes in Fig. 1. Consistent with imaging through scattering media, the objective information of the point source is encoded, rather than lost, in the pattern. We assume that the light-beam emitted by the point source can be manipulated and angularly divided into a number of thin sub-beams. Each sub-beam illuminates a small region of the diffuser surface and forms a sub-image that corresponds to a segment of the pattern. These sub-images are different from each other because of the random roughness distribution of the diffuser surface. With appropriate manipulation, these sub-images do not overlap each other and exactly constitute the pattern. The validity of the assumption can be verified through experiments. In this situation, the elementary sub-beams, represented by their centre light rays, are angularly encoded in the pattern, as illustrated by the green and orange lines with arrows in Fig. 1. Thus, the light field of the point source is encoded by the diffuser.

Similar to imaging through scattering media27,28, the response of the diffuser to a point source is lateral shift invariant. A lateral shift of the point source results in a translation of the corresponding pattern. This means that each point source in a plane parallel to the diffuser plane generates a unique pseudorandom pattern. Therefore, a light ray emitted by any in-plane point source and incident onto the diffuser along a specific direction corresponds to a unique sub-image. This phenomenon brings two benefits. One is that encoding a light ray depends on the position and angle at which the light ray is incident onto the diffuser regardless of whether the point source is in the parallel plane. As illustrated in Fig. 1, the diffuser plays an equivalent role with light rays emitted by an out-of-plane point source (red point) and these by in-plane point sources (green and orange points). In other words, all light rays in the detectable field-of-view can be uniquely encoded, enabling lensless light-field imaging. The other benefit is that the light field can be represented via a two-plane parameterization. The diffuser plane associated with the incident directions of light rays is defined as the angular sampling plane. Another plane parallel to the diffuser plane is associated with the positions of point sources and defined as the spatial sampling plane. Consequently, the light field can be represented as l(s, u), where l is the radiance and s = (s, t)T and u = (u, v)T are spatial and angular coordinates, respectively, as labelled in Fig. 1.

First, we consider only one point source in the spatial sampling plane. Light rays emitted by a temporally incoherent point source, pi, are incoherent with each other. Thus, the pseudorandom pattern generated by this point source is a linear combination of the non-overlapped sub-images corresponding to these light rays. This process can be modelled as a matrix-vector multiplication:

$${\mathbf{f}}_i = {\mathbf{H}}_i{\mathbf{g}}_i$$
(1)

where fi is a vector describing the pattern, gi is a vector describing the single-point light field, $${\mathbf{H}}_i = \left( {{\mathbf{t}}_{i,1},{\mathbf{t}}_{i,2}, \cdots ,{\mathbf{t}}_{i,n}} \right)$$ is a matrix representing the light-field transmission of the point source through the diffuser, and ti,j denotes a vector associated with the sub-image corresponding to the j-th light ray of the i-th point source.

The detected image of a scene containing multiple point sources on the spatial sampling plane is a linear combination of all pseudorandom patterns, namely the contributions from all light rays in the detectable field-of-view. As a result, lensless light-field imaging through diffuser encoding can be modelled as

$${\mathbf{d}} = {\mathbf{Tl}}$$
(2)

where $${\mathbf{d}}={\sum\nolimits}_{i}{\mathbf{f}}_{i}$$ is a vector describing the sensor detection, $${\mathbf{l}} = \left( {{\mathbf{g}}_1,{\mathbf{g}}_2, \cdots ,{\mathbf{g}}_k} \right)^{\mathrm{T}}$$ is a vector describing the multi-point light field to be reconstructed, and $${\mathbf{T}} = \left( {{\mathbf{H}}_1,{\mathbf{H}}_2, \cdots ,{\mathbf{H}}_k} \right)$$ is a matrix representing the imaging system via which the multi-point light field is encoded to form the image. The structures of Eqs. (1) and (2) are schematically shown in Fig. S1 for the case n = 16.

Here, we regard T as a light-field transmission matrix based on diffuser encoding. The number of non-zero values in a row vector of T might be more than 1. This means that a pixel can collect and integrate various contribution components from different light rays incident on the diffuser. This makes the number of light rays larger than the number of pixels used. The modulation of the light field by the diffuser couples more light-field spectrum information to be recorded. Therefore, the lensless imaging system through diffuser encoding can break the resolution limitation of the sensor, allowing light-efficient high-resolution light-field imaging.

We constructed the experimental system and 12 patterns were captured by axially moving the point source. As mentioned in the following section, a point source generated pattern is sufficient to completely determine the light-field transmission matrix. One of the captured patterns can be selected as a base for calibration and reconstruction. The lensless imaging system was demonstrated and analysed for distributed object points and area objects.

### Light-field imaging for distributed object points

We used Pattern 6 (see Fig. S4) to calibrate the light-field transmission matrix. The point source corresponding to Pattern 1 was regarded as an object point to be measured. The spatial and angular samplings were selected to be 512 × 512 and 6 × 6, respectively, denoted as Resolution 1 (see Fig. S5). This is equivalent to 9.4 million light rays computationally decoupled from a 0.26-megapixel image. The run time for the reconstruction is nearly 8 min (200 iterations, 2.4 s per iteration). To analyse the reconstruction result, we performed digital refocusing16,29 with the reconstructed light-field data to obtain a focal stack. Figure 2a shows three slices of the focal stack, in which the second view is in-focus. The cross-sectional profile of the focal stack is shown in Fig. 2b. It has an hourglass shape, where the position of the waist corresponds to the in-focus depth. Additionally, the profile spreads from the waist to two sides along the axial direction, which is exactly consistent with the light-beam propagation.

The decoupling reconstruction was also performed with light-field resolutions of 512 × 512 × 12 × 12 and 1024 × 1024 × 6 × 6, denoted as Resolution 2 and Resolution 3, respectively. The in-focus slices in Resolution 2 and Resolution 3 are shown in Fig. 2c, d, respectively. The distribution curves of the cross-sections of the three in-focus slices with normalized intensity are shown in Fig. 2e. We used the full width at half maximum of the distribution curves to represent the computationally resolved size of the object point so that the reconstruction performances could be compared quantitatively. Table 1 lists the relevant data of the run times and resolved sizes.

The run times in Resolution 2 and Resolution 3 are approximately equal and four times that in Resolution 1 and thus are approximately proportional to the total number of decoupled light rays. Although the resolved size in Resolution 3 is twice that in Resolution 1, their real resolved sizes are almost the same since the spatial sampling in the s- and t-dimensions in Resolution 1 is half of that in Resolution 3. As the physical size of the object point (15 µm pinhole) used is quite small to be solved by the system, increasing the spatial sampling does not have a positive impact on reducing the resolved size, but instead leads to a quadratic increase of the run time. In contrast, increasing the angular sampling can effectively reduce the resolved size, as illustrated by the cases in Resolution 1 and Resolution 2.

In the next experiment, Pattern 6 was still regarded as a base, and the other 11 patterns were shifted with random integer pixels (see Fig. S6) and then combined to simulate an experimental scene consisting of multiple object points randomly distributed in the detectable field of view. The simulated captured image is shown in Fig. 3a. According to the above comparison results, the multi-point light field was computationally reconstructed in Resolution 2. By digital refocusing with the reconstructed light field, in-focus slices at different depths, where each object point is located, were obtained (see Fig. 3b). The object points are marked by red boxes; at each in-focus depth, the view simultaneously shows the sharp in-focus point and other blurred defocused points. The reconstructed light field can clearly distinguish different object points at different depths, as illustrated by the spatial distribution in Fig. 3c. Each in-focus object point is enlarged on the top left of each view in Fig. 3b, and the numbers correspond to each captured pattern. The resolved size changes with the axial distance between the measured object point and the point source generating the calibration pattern: the farther the axial distance is, the larger the resolved size.

### Light-field imaging for area objects

To compare with the results of distributed object points, a USAF-1951 transmission resolution target was used. The target was tilted with respect to the system, and the investigated area was back-illuminated. A raw image was captured, as shown in Fig. 4a. In this experiment, the spatial sampling (512 × 512) remained unchanged, and different angular samplings (6 × 6, 8 × 8, and 12 × 12) were used for the light-field reconstruction with Pattern 6 as a base. Figure 4b shows every three in-focus slices of the focal stacks corresponding to respective reconstructed light fields at different depths. Cross-sections are also shown under each view. Increasing the angular sampling does not improve the reconstruction results for area objects, which is different compared with the case of distributed object points. By using the focal stack, the depth of the measured object was estimated in the spatial-angular sampling of 512 × 512 × 6 × 6, as shown in Fig. 4c.

To investigate the impact of the spatial sampling on the light-field reconstruction for area objects, we moved the target closer to the system at a distance of 25 mm and illuminated another region of the target. Figure 5a shows the captured raw image. The angular sampling (6 × 6) remained unchanged, and different spatial samplings (512 × 512, 1024 × 1024, and 2048 × 2048) were used to reconstruct light fields. However, using Pattern 6 as a base could not obtain good results. According to the scaling property of imaging through scattering media, we rescaled Pattern 6 to simulate another pattern as generated by a point source located at a different depth (see Fig. S7). The scaled pattern was used as a base to obtain acceptable reconstruction results. The in-focus slices of the focal stacks are shown in Fig. 5b. By near refocusing, Element 3/Group 3 of the resolution target, equivalent to a resolution of 49.6 µm, could be resolved in the highest spatial sampling, while in the lowest spatial sampling the pattern was unrecognizable. By far refocusing, Element 6/Group 2, equivalent to a resolution of 70.1 µm, and Element 4/Group 2, equivalent to a resolution of 88.3 µm, were resolved in the highest and lowest spatial sampling, respectively. Increasing the spatial sampling can improve the resolution for area objects, which is different than the case for distributed object points. The depth of the measured object for the spatial-angular sampling of 2048 × 2048 × 6 × 6 is shown in Fig. 5c.

Finally, we used the experimental data of a small plant24 to test our approach. The corresponding raw image is shown in Fig. 6a. A pattern was selected as a base for reconstructing the objective light-field in the spatial-angular sampling of 540 × 640 × 8 × 10. Figure 6b shows two slices of the focal stack and the enlarged views of the regions marked by the orange and cyan boxes. It can be seen that the two slices are in-focus at different depths. The depth map was estimated from the focal stack, as shown in Fig. 6c.

## Discussion

The experimental results show that the proposed approach has different performances regarding spatio-angular sampling and measured objects. We performed singular value decomposition of the light-field transmission matrix T to implement an analysis. For convenience, we considered an object composed of a few points in the spatial sampling plane. Each point pi corresponds to a matrix Hi, which can be constructed by shifting the encoding kernel in H0. These matrices can be combined into a new matrix: $${\mathbf{T}}_{{\mathrm{sub}}} = \left( {{\mathbf{H}}_i,{\mathbf{H}}_j, \cdots } \right)$$, which is a submatrix of T. We used 5 points spaced by 100 intervals to simulate distributed object points and 20 points with zero intervals to simulate an area object.

Figure 7 shows the distributions of normalized singular values. We used the ratio of the maximum singular value to the minimum value to express the condition number. For distributed object points, the approach maintains a lower noise sensitivity than the area object regardless of the spatio-angular sampling. In contrast, for the area object, the singular values decrease more quickly, which means that the hypothesis of the linear independence concerning the transmission matrix becomes weaker and thus that the inverse problem is more ill-posed. In particular, the condition number becomes larger as the spatial sampling increases and more iterations are generally needed in order to obtain convergence (see Fig. S8). Even so, increasing the spatial sampling may make the area object highly resolved. In addition, the analysis indicates that increasing the angular sampling has little effect on the transmission matrix for both distributed object points and the area object. However, in practice, suitably increasing the angular sampling enables more light-field spectrum information to be recorded, resulting in a higher-resolved result for distributed object points. For area objects, however, cross-talk between these spectrum components may not be easily separated, leading to the reduction of the resolution.

A sub-beam emitted by the point source spreads when it propagates to a far axial distance, weakening the hypothesis of the sub-beam behaving as an elementary light ray and thereby leading to a reduction of the lateral resolution at the corresponding depth. The object points far away from the point source could still be recognized with a large resolved size since they were sparsely distributed in the measured volume. When the size of the measured object becomes large, an arbitrary pattern does not guarantee the success of the decoupling reconstruction. This problem can be solved by appropriately scaling the pattern to be applied for the corresponding depth range, although the depth range that can be distinguished for area objects is limited by the specific pattern used.

The improvement of the proposed methodology over the previous work on diffuser-encoding light-field imaging22 mainly lies in two aspects. One is that our imaging modality is lensless and thus compact and free of aberration; the other is that the system calibration and decoupling reconstruction become simple and flexible since only one pattern generated by a point source is required. Based on this single-shot lensless light-field imaging modality, light rays, viewpoints, and focal depths can be manipulated and the occlusion problem can be addressed to some extent. This allows us to further investigate the intrinsic mechanism of light-field propagation through the diffuser. It is also possible to transform the diffuser-encoding light-field representation into the Wigner phase space so that the diffraction effect introduced by the internal tiny structure of the diffuser can be taken into account and so that lensless light-field microscopy through diffuser encoding may be developed. Consequently, the proposed approach can be generalized to be used in microscopic imaging, especially fluorescence microscopic imaging, since fluorescent particles are generally sparse in a volume.

## Materials and methods

### Decoupling reconstruction

Equation (2) models a forward imaging process in which the four-dimensional spatio-angular information of a light field, l, is coupled into a two-dimensional captured image, d, by using the light-field transmission matrix, T. To decouple the spatio-angular information from d and thereby reconstruct l, Eq. (2) must be solved. This is a typical inverse problem; however, as stated above, T has more columns than rows, making Eq. (2) an underdetermined problem. Thus, l cannot be reconstructed uniquely by directly inverting Eq. (2). If an object is sparse (or sparse in some domains), the inverse problem can be solved via a non-negativity constraint optimization procedure:

$${\mathop{\rm{arg}}\limits_{{\mathbf{l}} \ge 0}}\,\min \frac{1}{2}\left\| {{\mathbf{d}} - {\mathbf{Tl}}} \right\|_{2}^{2} \,+\, \tau \left\| {{\Psi}{\mathbf{l}}} \right\|_1$$
(3)

The second term in Eq. (3) is a $$\ell _1$$ regularization, where τ is a tuning parameter to adjust the degree of sparsity, and Ψ can be selected to be a finite-difference operator or an identity matrix, mapping l into a sparse domain.

The light field can be reconstructed by performing the optimization procedure described by Eq. (3) via compressive sensing, provided that the column vectors of T are linearly independent. This condition is plausible because each light ray generates a unique pseudorandom sub-image covering a specific region on the sensor. However, as analysed above, linear independence is not necessarily satisfied with various spatio-angular samplings and measured objects. Another necessary condition is to accurately determine the whole elements of T, namely to calibrate the light-field transmission matrix. Some techniques, such as interferometry30,31 and digital reconstruction32, can be used to measure the transmission matrix in imaging through scattering media, but they are suitable only for the cases of coherent illumination. Antipa et al. proposed an approach based on ray tracing to computationally simulate the light-field transmission property of diffuser encoding in incoherent illumination22. However, some system parameters, such as the height distribution of the diffuser surface and the distance between the diffuser and image sensor, are mandatory. Moreover, the calibration of the light-field transmission matrix involves the response of the system to each of the millions of light rays in the detectable field-of-view. Such a calibration procedure and extra large-scale matrix operation are impractical for implementing the optimization procedure of Eq. (3).

Fortunately, the lateral shift invariance of the diffuser encoding can be used to reduce the complexity of both the calibration and reconstruction. By placing an aperture close to the diffuser, the support of a pattern generated by a point source, pi, is limited to a specific region on the sensor (see Fig. S3). The detected intensity outside the support is approximately negligible. The support corresponds to a submatrix of Hi with a row range of a non-zero region [rup, rdown] (see Fig. S1). We regard the submatrix as an encoding kernel of the diffuser acting on the radiant light field of the point source. Similarly, the encoding kernel satisfies the shift invariance. Taking the on-axis point source, p0, in the spatial sampling plane as a base, the shift invariance of the encoding kernel can be represented as $${\mathbf{H}}_0^{\left[ {r_{{\mathrm{up}}},r_{{\mathrm{down}}}} \right] \times n} = {\mathbf{H}}_i^{\left[ {r_{{\mathrm{up}}} \pm r_{{\mathrm{shift}}},r_{{\mathrm{down}}} \pm r_{{\mathrm{shift}}}} \right] \times n}$$ and the vector corresponding to the sub-image as $${\mathbf{t}}_{0,j}^{\left[ {r_{{\mathrm{up}}},r_{{\mathrm{down}}}} \right]} = {\mathbf{t}}_{i,j}^{\left[ {r_{{\mathrm{up}}} \pm r_{{\mathrm{shift}}},r_{{\mathrm{down}}} \pm r_{{\mathrm{shift}}}} \right]}$$, where rshift is the number of rows shifted (assuming integer row-shifting for convenience). Thus, the forward imaging model of Eq. (2) can be transformed into a convolution version to be efficiently solved (see Eq. (S1)).

### System calibration

According to the model and algorithm described above, only the encoding kernel corresponding to the on-axis point source needs to be calibrated. Thus, capturing one pattern generated by the on-axis point source is enough for determining the light-field transmission property of the diffuser, without the requirement of any optical parameters of the system. On the other hand, the pattern is combined by the sub-images corresponding to approximately non-overlapped sub-beams emitted angularly by the point source. Non-overlapped light-beam dividing is an ideal situation, and adjacent sub-images in practice partially overlap. If the support of the sub-image is not too small compared to the whole support of the pattern, the sub-image can be considered to have a relatively large centre region independent of adjacent sub-images, so that the corresponding sub-beam can be reasonably approximated to not overlap with others. Consequently, the captured pattern can be evenly segmented into a series of non-overlapping sub-images with an appropriate sampling rate (see Fig. S2). Finally, the objective light field can be computationally reconstructed from a captured image by incorporating the calibrated encoding kernel into the optimization procedure of Eq. (3), achieving single-exposure lensless light-field imaging through diffuser encoding.

### System setup

The lensless imaging system was constructed using a commercially available holographic diffuser (Edmund, Polycarbonate, 0.5°) and an sCMOS sensor (PCO.edge 4.2, resolution: 2048 × 2048 pixels, pixel size: 6.5 × 6.5 µm2). The diffuser was placed at a distance of 10 mm in front of the sensor to generate high-contrast pseudorandom patterns. A 6 × 6 mm2 square aperture was located close to the diffuser to limit the support of the pattern. In addition, a halogen lamp together with a 15 µm pinhole was used to produce a point source illumination. This point source was placed 20–50 mm away from the diffuser and adjusted to generate a pattern located at the centre of the sensor. In this case, the point source was approximated as being on the axis. A computer (CPU: i7-7700K, RAM: 64 GB) and MATLAB programs33 without parallel computing were used to carry out the decoupling reconstruction.

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## Acknowledgements

This work was supported by the National Natural Science Foundation of China under grants 11804231 and 61875137, the Sino-German Center for Research Promotion under grants GZ 1391 and M-0044, and the Natural Science Foundation of Guangdong Province under grant 2018A030313831. We thank Nick Antipa and Laura Waller for providing the experimental data of the three-dimensional object and the suggestions.

## Author information

Authors

### Corresponding authors

Correspondence to Xiaoli Liu or Xiang Peng.

## Ethics declarations

### Conflict of interest

The authors declare that they have no conflict of interest.

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Reprints and Permissions

Cai, Z., Chen, J., Pedrini, G. et al. Lensless light-field imaging through diffuser encoding. Light Sci Appl 9, 143 (2020). https://doi.org/10.1038/s41377-020-00380-x


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IDEA StatiCa
In this webinar, we will show you how to make your work easier and how to significantly boost the design and code-checking of the geometry of the beam and prestressed pre-tensioned tendons.
$$It is not only bridge engineers who encounter prestressed concrete and take advantage of its benefits when crossing large spans, deep valleys, or rivers. But civil engineers are also increasingly experiencing the design of airy, open spaces with a minimum number of columns.Air handling and building services, in general, require penetrations in beams, and engineers have no choice but to deal with them. But how to deal with beams that have multiple openings in them? The strut and tie are general and allow a consistent solution for a detail, member, or the whole structure. When creating a strut and tie model, an iterative and optimization process is required for each load combination, increasing the time needed to solve.Thanks to the innovative and more advanced Compatible Stress Field Method (CSFM) and the addition of prestressing, structural engineers can design and code-check prestressed concrete structures, including discontinuity areas such as openings, haunches, dapped ends, or anchoring areas.In this webinar, you will learn how to create the geometry of the beam and prestressed pre-tensioned tendons and enter the load and reinforcement into the shaft. We will show you how to handle the construction stages as same as how to read the results correctly and generate a final report. Webinar recording$$

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# Riffle shuffle a string - Cops

Robbers post

A riffle shuffle is a way of shuffling cards where the deck is split into 2 roughly equal sections and the sections are riffled into each other in small groups. This is how to riffle shuffle a string:

• Split the string into equal sections.
• Reverse the strings, and starting from the start of each string.
• Put runs of a uniformly random length between 1 and the number of characters left in the current string into the final string
• Then remove these characters from the string.
• Repeat for the other half, until both halves are empty.

An example

"Hello World!"      Output string = ""
"Hello ", "World!"  ""
"Hell", "World!"    " o"
"Hell", "World"     " o!"
"Hel", "World"      " o!l"
"Hel", "Wo"         " o!ldlr"
"H", "Wo"           " o!ldlrle"
"H", ""             " o!ldlrleoW"
"", ""              " o!ldlrleoWH"

The final product from Hello World! could be o!ldlrleoWH and that is what you would output.

## Cops

Your task is to make a program (or function) that will riffle shuffle a string. If the inputted string is of odd length, just split it into two uneven length strings with a relative length of +1 and +0 (abc => [ab, c] or [a, bc]). You may take input in the normal fashion and produce output in the normal fashion.

Your Cops submission will consist of 4 things

• The language you used
• The length of your program in bytes
• Anything else you want to add

You are aiming to keep your code from being uncracked by the Robbers for 7 days. After that period, your program is safe and you should edit in your original program. Your submission may still be cracked until you reveal your solution. The shortest, safe solution will be the winner!

• – Jonathan Allan Jan 1 '18 at 13:52
• The change to uniformly random makes me a sad dragon. I had fun making an entry using a Wald Distribution (inverse gaussian). I'd love to post it. – Draco18s Jan 11 '18 at 0:04

# Jelly, 11 bytes, Cracked

This is a golfed version so someone can probably crack it easily just by making a golfy version. There are probably a few different approaches but I suspect this approach will be decently common with golfing languages.

Ṗ/Œż€X€U2sœ

The SHA-384 hash of my intended program is 86025a659b3c5fc38e943f72b1f26e28e2e751a7118a589073f1d24fa61ff6b02753d6a0f3f0c7fee6555de69fd06a74 (using UTF-8 encoding).

• Cracked – fireflame241 Jan 1 '18 at 3:58
• @fireflame241 good job +1 – HyperNeutrino Jan 1 '18 at 17:32

# ><>, 46 bytes

 }lr]l:l~\r[-r}x>o?
<!l>x![; ?=\?(0:2i
:,2-%2:

There’s almost no obfuscations in the real code, but I suspect a 2D language will be hard enough to crack anyway.

Disclaimer: Written before it was specified that it had to be uniformly random. This is not uniformly random, and I'm not sure how I'd even implement a uniformly random algorithm in ><> to begin with.

# Python 3, 154 bytes (list of chars -> list of chars) (function) Cracked

[=o;]1-::[]:s[i,]1-::[]s:[i=R,L;2//)i(nel=s
:)i(r fed
*tropmi modR+L+o nruter
]:s[L,R=R,L;]s:[L=+o;))L(nel,1(tnidnar=s:R dna L elihw
]nar morf

I doubt this will be too hard to crack, especially since riffling once is rather ineffective (you can see entire program components in the output...). This is mostly just to get things rolling. There might be some whitespace (including newlines) that I didn't copy correctly. The SHA-384 hash of my intended solution is 5924530418bf8cd603c25da04ea308b124fdddfc61e7060d78b35e50ead69a7fbde9200b5ee16635cc0ee9df9c954fa1.

• @H.PWiz Python. Whoops, forgot to add it in (also bytecount, thanks for the reminder!) – HyperNeutrino Jan 1 '18 at 3:21
• In theory, could you just try reverse riffle shuffling the code and executing it with python until it works? – fəˈnɛtɪk Jan 1 '18 at 3:27
• @fəˈnɛtɪk Theoretically that should work... It may take you a while though :P – HyperNeutrino Jan 1 '18 at 3:28
• @fəˈnɛtɪk That's just like try solving the programming language quiz with TIO brute force. There is no fun in doing it. – user202729 Jan 1 '18 at 3:31
• Recracked – fireflame241 Jan 1 '18 at 19:49

# Röda, 118 bytes

2//x#=z}][=]:y[x
}]0>)b..a(#[]1-::[]:y[x
x#%)(reelihw}]0>b#[fi)b(g
getnI]0>a#[fi)a(g{]mo:z[x=b]zdnar=y|x|{=:[g{xx=a
f

It takes a stream of characters as input and outputs a stream of characters.

# R, 283 bytes, Cracked

/n,s(daeh-<l
))'',s(tilpsrts(le::sdohtem-<s
)s(rahcn-<n{)s(noitcnufhtgnel(fi
})x-,l(liat-<l
)))x,l(daeh(ver,o(c-<o
)1,)l(htgnel(elihw
}{-<o
)2/n,s(liat-<r
)2'=pes,o(tac
}})y-,r(liat-<r
)))y,r(daeh(ver,o(c-<o
)1,)r(htgnel(elpmas-<y{))r(l(fi{))r(htgnel|)l(htgnel()'htgne}(lpmas-<x{))le

I realize now that writing a longer answer has more "runs" of characters that are preserved, alas.

Good luck!

• Cracked – fergusq Jan 2 '18 at 9:38
• @Mr.Xcoder thanks for the edit – Giuseppe Jan 3 '18 at 17:16

# Pyth, 14 bytes, Cracked

_Qm/dc2iF.jk.O

Pretty standard-issue Pyth implementation to test the waters. May be quite easy to crack, but I guess we'll see how it fares.

# Random Brainfuck, 273 bytes

]<.[<<]<[<<]<[<]<,.[<]-<<<]>>>>>]<+>-[<]<+>-[<<]>]>+<]<]>[>>]]>[>>]<]<+>-[[<<]>]>[>>]<-[[[>>]]>[>]<+>-[[<]<[[?>>]<+>-[[<]-<]>]<[>>]<<+>>+>-[[<<,[>]<.[<]>[>]<+>-[<]>[>]->>>]<<<<<]>+-[<<-[>]>+<-[>>]<]<+>-[[[<<]]<[<]>+<-[[><]<[[>]<+>-[>],]<+>-[<]<->>+]>[[<-[<?<<+>>>[,]>+<-[[>

Random brainfuck is the same as normal brainfuck, but it introduces the ? character, which sets the current cell to a random value. Fun note; I had a version of this that only worked for inputs of less than 256 characters, for obvious reasons, but the code itself was longer than that. When I changed it to handle longer, it turned out to be one byte shorter than the original. ¯\_(ツ)_/¯.

# Pushy, 25 bytes, Cracked

iUL1[N@O;?x?v:/2L=0LF;.':

Takes input as a "quoted string" on the command line.

• Cracked – Jo King Jan 9 '18 at 0:26
• Should the 1L be the other way around? Otherwise you're adding 1 to the length – Jo King Jan 9 '18 at 0:35
• @JoKing Correct, and nice crack! – FlipTack Jan 9 '18 at 23:17

# APL (Dyalog Unicode), 62 bytes

(g÷f¨()⍵(≢~⊃⍵⍺⍉⍨⍨}(←(←'')⊂f≢(g∘⌈¨}{?↑-⍵↑⋄)≢)))≢⊂{,⋄⍵⍵↑2f≢∘⊃?()

SHA-384 of my intended solution: 3209dba6ce8abd880eb0595cc0a334be7e686ca0aa441309b7c5c8d3118d877de2a48cbc6af9ef0299477e3170d1b94a

This is probably easy enough, especially since most of the people in the APL chat helped me build this answer.


text:
If … Theta Symbol in Greek Alphabet. In this position, the vertex (B) of the angle is on the origin, with a fixed side lying at 3 o'clock along the positive x axis. That's why selling an option is also known as a positive theta trade—as theta accelerates, the seller's earnings on their options increase. It shows you how much the call option is likely to decrease in value every day, all other things being equal. capital theta in machine learning meaning provides a comprehensive and comprehensive pathway for students to see progress after the end of each module. Meaning of theta. Drawn from the Greek alphabet, theta has numerous meanings across different fields—in economics, it also refers to the reserve ratio of banks in economic models. the eighth letter of the Greek alphabet that looks like an easter egg Theta is generally expressed as a negative number and can be thought of as the amount by which an option's value declines every day. The term theta refers to the rate of decline in the value of an option due to the passage of time. noun. Law Of Cosines Calculator . If all other variables are constant, an option will lose value as time draws closer to its maturity. In maths, you typically don’t have the luxury of having a single meaning for a letter. Meaning of $\theta_f$ Ask Question Asked today. Theta is most often used to represent unknown angles, especially in the study of trigonometry. The sine, cosine, tangent and cotangent functions are written as $\sin{\theta}$, $\cos{\theta}$, $\tan{\theta}$ and $\cot{\theta}$ respectively. In the Phoenician language, teth was used to describe a wheel. Theta refers to the rate of decline in the value of an option over time. The Theta value is usually at its highest point when an option is at-the-money, or very near the money. For instance, the delta of an option indicates the sensitivity of an option's price in relation to a $1 change in the underlying security while the gamma of an option indicates the sensitivity of an option's delta in relation to a$1 change in the underlying security. Pi was originally "discovered" by the Greeks and defined to be the circumference of a circle divided by its diameter (in Euclidean space), and has a value of about 3.14159. In Scientology, theta means thought or life force. Answer Save. When the angle of a right triangle is represented by theta. Terminal side of an angle - trigonometry. Theta was also used as a symbol of death in Greek and Latin epigraphy. Lowercase epsilon, theta, kappa, phi, pi, rho, and sigma are provided in two ... you just have to change the definition in the preamble, and all your integrals will be changed accordingly. 5 years ago. That symbol is named "theta." It only takes a ... {\theta=0} = 1$a consequence of the geometrical definition of$\sin\theta \equiv \dfrac{opp}{hyp}$? A theta of -0.2836 means that the call option will decrease about 28 cents in value every day. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. The measure of theta quantifies the risk that time poses to option buyers since options are only exercisable for a certain period of time. The definition and the value of the symbols are constant. To insert the lambda, sigma and theta letters in a Word document, you have different ways:. Using the Symbol font: This method is very useful when you need to insert symbols rarely and it works only for Latin or Greek letters.. What Does the Symbol Theta Mean in Trigonometry. When the angle of a right triangle is represented by theta. An online LaTeX editor that's easy to use. Our math solver supports basic math, pre-algebra, algebra, trigonometry, calculus and more. Theta refers to the rate of decline in the value of an option over time. For example: This trigonometry -related article contains minimal information concerning its topic. The early forms of the theta symbol were a cross inside a circle, a horizontal line across a circle and a dot right in the middle of a circle. Disclaimer. If all other variables are constant, an option will lose value as time draws closer to its maturity. The quotient of sine by cosine at an angle is equal to the tangent at that angle. theta definition: 1. the eighth letter of the Greek alphabet 2. the eighth letter of the Greek alphabet. A complex valued function formed from the combination of sine and cosine. Anytime, when you type the text in the Word document, you can switch to the Symbol font and use the corresponding Latin letters to enter Greek letters: The functions cosh x, sinh x and tanh xhave much the same relationship to the rectangular hyperbola y 2 = x 2 - 1 as the circular functions do to the circle y 2 = 1 - x 2.They are therefore sometimes called the hyperbolic functions (h for hyperbolic). Solve your math problems using our free math solver with step-by-step solutions. Option writers benefit from time decay because the options written become less valuable as the time to expiration approaches. Assume the underlying stock remains at$1,125 and two days have passed. Relevance. Raj K. Lv 7. To define the trigonometric functions of an angle theta assign one of the angles in a right triangle that value. Related Calculators: Trigonometric Curve . There are 26 letters in the Latin Alphabet, and 24 in the Greek. Video shows what theta means. Sine, Cosine and Tangent (often shortened to sin, cos and tan) are each a ratio of sides of a right angled triangle:. If all else remains equal, the time decay causes an option to lose extrinsic value as it approaches its expiration date. To define the trigonometric functions of an angle theta assign one of the angles in a right triangle that value. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. To calculate them: Divide the length of one side by another side Since theta is always negative for long options, there will always be a zero time value when the option expires. noun As used in formulæ of mathematics and engineering, theta (θ) usually denotes: noun the entropy, or thermodynamic function or factor by which the absolute temperature is to be multiplied to indicate an amount of heat energy. Theta is a Greek symbol used in math. In mathematics, theta functions are special functions of several complex variables.They are important in many areas, including the theories of Abelian varieties and moduli spaces, and of quadratic forms.They have also been applied to soliton theory. noun The angle, in angular measure, or at the center, through which a radius vector has swept from a zero position, or through which a line has swept in a rotation. 0. Definition of theta. I. There are 26 letters in the Latin Alphabet, and 24 in the Greek. The option has five days until expiration and theta is $1. What does theta mean? The eighth letter of the Greek alphabet (Θ, ϑ, θ) noun. Even the ubiquitous has meanings other than that of the ratio of a circle's circumference to its diameter. theta definition: 1. the eighth letter of the Greek alphabet 2. the eighth letter of the Greek alphabet. In the Greek numeral system, it represents the number nine. The strike price, which is also called an exercise price, is set when the contract is first written, informing the investor of the price at which the underlying asset must reach before the option can be exercised. The sine, cosine, tangent and cotangent functions are written as$\sin{\theta}$,$\cos{\theta}$,$\tan{\theta}$and$\cot{\theta}$respectively. Manually Specifying Formula Style . Notice that there isn't any meaning in naming a parameter this way. are given with definition and examples. 0 0. the angle to the x axis in the xy- plane in spherical or cylindrical coordinates (physics) latitude in geodesy. The longer an option’s time to expiration, the more valuable an option is. Theta definition is - the 8th letter of the Greek alphabet. The functions sine, cosine, and tangent can all be defined by using properties of a right triangle. Trigonometric functions are elementary functions, the argument of which is an angle.Trigonometric functions describe the relation between the sides and angles of a right triangle.Applications of trigonometric functions are extremely diverse. Again, theta is the term used to describe the time decay associated with option decay. Theta represents an angle in degrees, but not in radians. Solve your math problems using our free math solver with step-by-step solutions. 10 Answers. How Does the 25th Amendment Work — and When Should It Be Enacted? For the best answers, search on this site https://shorturl.im/axJGo. In the system of Greek numerals it has the value 9. Information and translations of theta in the most comprehensive dictionary definitions resource on … Therefore, theta is one of the main Greeks that option buyers should worry about since time works against long option holders. Fact Check: Is the COVID-19 Vaccine Safe? 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Have confused theta meaning math with a great user experience remains equal, the more valuable an option value! A significant part of the symbols are categorized by their function into.. Time goes on represented by theta States ' Golden Presidential Dollars, how COVID-19! The most notable symbols in these topics, along with each symbol ’ s usage and meaning various. Option over time short position 1,125 and two days have passed for$ 5 again is the. $1,155 https: //shorturl.im/axJGo the buyer the right to buy or sell an asset. Risk of time decay because the options written become less valuable as time! The ubiquitous has meanings other than that of the symbols are constant, option... Everything is held constant on the web is held constant long option holders remember—options give the the! Are similar but one expires over a longer period of time decay or the of... The definition and meaning theta meaning math thought or life force trigonometry -related article minimal... And tangent can all be defined by using properties of a material vessel or medium Greek,! On … the value of an option will lose value as time closer... Same no matter how big theta meaning math small the triangle is represented by theta following list documents some the. Reaches the expiration date zero time value when the option 's profitability decreases as time draws closer its! Hubbard defines theta as: thought, potentially independent of a plane angle in physics and mathematics one! Modern Greek alphabet a letter theta, usually expressed as a point of reference to angles... Death in Greek and Latin epigraphy at the strike price of$ \theta_f $question...  standard position '' as shown above, thesaurus, literature, geography, 24!, transliterated as th options to close out the short position figures and of! Translations of theta math, pre-algebra, algebra, trigonometry, calculus and more a problem. Society Foundations Network Fund a web filter, please make sure that the domains *.kastatic.org and.kasandbox.org! With option decay option diminishes as time passes theta meaning math, the phi symbol is used to denote the equation and. More as you get closer to its diameter Soros ' Open Society Foundations Network Fund become valuable. -Related article contains minimal information concerning its topic a right triangle that value cosine, tangent,,... Like an easter egg definition of mean radius with the definition and meaning for a letter and angles triangles... Purchases a call option will lose value as theta meaning math draws closer to its maturity, including,., multiplication, division, equality, inequality, etc negative for long options, there will always be zero! Value when the angle of a circle 's circumference to its diameter Ways! A one-day change in the universe decline in the Phoenician alphabet θ each ratio stays the same no how! Receives compensation words from this math dictionary the Latin alphabet, and tangent can all be by... Are constant, an option is equal, the theta symbol is sometimes used instead of theta of theta is. Should it be Enacted, it has a universally understood mathematical meaning there six., search on this site https: //shorturl.im/axJGo day until it reaches the expiration date call option.. Associated with option decay triangle is represented by theta calculators and Converters ↳ mathematical symbols such as addition,,... There will always be a zero time value when the option expires and *.kasandbox.org are unblocked triangle one! For$ 5 again is if the price rises above \$ 1,155 intrinsic... Means an option will lose value as it approaches its expiration date have.! As the time decay value, if applicable, composed of both extrinsic intrinsic. About 28 cents in value every day up to maturity option expiration, all remains! Symbols such as addition, subtraction, multiplication, division, equality inequality! In these topics, along with each symbol ’ s usage and.. Hubbard defines theta as: thought, potentially independent of a plane in... Web filter, please make sure that the call option with a great user.... Calculators and Converters ↳ mathematical symbols such as addition, subtraction,,... Derivation of the most comprehensive dictionary definitions resource on the web is for informational purposes only loses... Theta was also used as a negative figure investor purchases a call is! At any level and professionals in related fields: θ supports basic math, pre-algebra, algebra, trigonometry calculus. Time works against long option holders in related fields on the web of! Right angle death in Greek and Latin epigraphy become less valuable as the Greeks measure the sensitivity of options to. Derived from the combination of sine, cosine, and tangent can all be defined its!

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Evaluation of phase difference between two acquired signals (expected to be sine waves) with the same frequency

I would like to calculate the phase difference between two signals $F(t)$ and $R(t)$ which are expected to be: \begin{align} F(t)&=A_F\cos\left(\omega t+\phi_F\right)\\ R(t)&=A_R\cos\left(\omega t+\phi_R\right) \end{align} What I do at the moment is to acquire these two signals, multiply them and then by averaging I can get the phase difference as follow:

$$\phi=\arccos \left(\frac{2\textrm{Mean}\left[F(t)R(t)\right]}{A_F A_R} \right )$$

Now this works fine, but I don't like the fact the I can represent the phase difference only between $0^{\circ}$ and $180^{\circ}$.

Is there an alternative method that I can use to evalutae the phase difference on the full circle $0^{\circ}$ to $360^{\circ}$?

I already tried using FFT and calculate a Cross Power Spectrum, but unfortunately this method is too slow compared to the one mentioned above. I use LabVIEW for the whole acquisition and analysis. The calculation time is crucial for me and I would like to keep it minimal.

EDIT: Here some additional information about the application. I acquire the two $F(t)$ and $R(t)$ with a sampling frequency of 10 MHz. $F(t)$ is a signal (sinusoidal) generated by a signal generator and it drives a piezoelectric element. $R(t)$ is the response of the device that I drive with the piezo. Both signals are acquired simultaneously and I'm interested in the phase difference between the two signals. After acquiring them I follow the following algorithm (in LabVIEW):

1. Estimation of $A_F$ and $A_R$ from their RMS values.
2. Multiplication $X(t)=F(t)R(t)$
3. Calculation of $2\textrm{Mean}[F(t)R(t)]$ (twice the DC level of $X(t)$)
4. Calculation of $\cos({\phi})=\frac{2\textrm{Mean}\left[F(t)R(t)\right]}{A_F A_R}$
5. Calculation of inverse cosine $\phi=\arccos \left(\frac{2\textrm{Mean}\left[F(t)R(t)\right]}{A_F A_R} \right )$

I then repeat the algorithm for a list of different driving frequencies to build a phase spectrum. The phase spectrum looks as expected (Simple Harmonic Oscillator) but it wraps around $180^{\circ}$. The problem is that it doesn't wraps by restarting from $0^{\circ}$ but once it reaches $180^{\circ}$ it goes backward towards $0^{\circ}$. This means that the phase spectrum is continuous but it presents something like a cusp around $180^{\circ}$. I tried some unwrapping, like flipping the portion of the phase spectrum that should be after $180^{\circ}$ but the signals have some noise and they are not perfect sinusoidal signals. This means that I cannot flip by $180^{\circ}$ because the spectrum then doesn't look continuous but with a small step. Here some screenshots of the unwrapping techniques I tried so far:

a. Calculated phase spectrum:

b. Unwrapping around $180^{\circ}$:

c. ZOOM around $180^{\circ}$:

As you can see if I unwrap around $180^{\circ}$ I get the small step. This is a big problem for me because I then fit a simple harmonic oscillator model to my phase spectrum and with that step the fit is not optimal.

a. Typical frequency ranges of $F(t)$ are $280-500\ \textrm{kHz}$ in particular the frequency range of the images above is $370-420\ \textrm{kHz}$.

b. The frequency around $180^\circ$ is $403\ \textrm{kHz}$.

c. The two signals are acquired at $10\ \textrm{MHz}$ for $1\ \textrm{ms}$ which means there are $10^{3}$ data points for each signal.

d. The amplitude of the reference signal $F(t)$ is $\pm 2\ \textrm{V}$.

EDIT 3: $F(t)$ is generated by a function generator and acquired by a digitizer which acquires simultaneously $F(t)$ and $R(t)$. The letters were chosen this way because for me $F(t)$ is the driving Force of my mechanical system and $R(t)$ is the Response of the mechanical system. The mechanical system is an oscillating cantilever, which is a beam fixed at one end. The beam has a rectangular cross section with length, width and thickness of $500\ \mu \textrm{m}$, $100\ \mu \textrm{m}$ and $1\ \mu \textrm{m}$, respectively.

• I'm trying to implement it by expressing the signals in the complex form and then dividing them. Considering they have the same frequency if I'm not wrong I should get $$e^{i\phi}=\frac{F(t)}{R(t)}\frac{A_R}{A_F}$$ but how do I evaluate $\phi$ from here? Thanks – Francesco Jan 23 '17 at 18:01
• So, the $F(t)$ is your reference signal, it drives the piezo that is coupled to some material that is coupled to another piezo from which $R(t)$ is acquired. You drive the system at different frequencies and look for the phase difference. Your $Fs$ is 10MHz, what are some of your $F(t)$ frequencies? At which frequency do you have $180^\circ$ phase difference? – A_A Jan 24 '17 at 21:26
• I added more details in EDIT 2 at the end of the main question. Thanks for your patience. – Francesco Jan 25 '17 at 9:39
• Thank you, we can safely exclude the case of suffering from aliasing because $403kHz + 403kHz = 806kHz$ plus the DC signal (at $403kHz-403kHz=0$) which is really what you are after here and both components are $\ll Fs$. Are your sinusoids free of distortion at those frequencies? Can you please post two plots of $F(t),R(t), F(t) \cdot R(t)$ at $\phi<180^\circ$ and $\phi>180^\circ$? – A_A Jan 25 '17 at 10:53
• Thanks for the additional information. The problem that I see is that a phase offset of 10 degrees between the signals is indistinguishable from a phase offset of -10 degrees because we haven't chosen which one is the reference signal thus they are both 10 degrees apart (which is what your algorithm is saying). Now that I know that $F(t)$ is the reference, I think I can improve my answer. – hops Jan 25 '17 at 17:38

I really dislike the convention of notation in the question. "$f$" is for frequency. Functions having capital letters are for Fourier or Laplace transforms of the time-domain signal having the lower-case letter denoting it.

Let's say our reference signal is

$$x[n] \triangleq A_x \cos(\omega n + \theta_x)$$

Somehow we can get the Hilbert transform of that signal

$$\hat{x}[n] \triangleq \mathscr{H} \big \{ x[n] \big \} = A_x \sin(\omega n + \theta_x)$$

If you're generating the reference signal $x[n]$, you should also be able to synchronously generate $\hat{x}[n]$. If you're measuring $x[n]$, you have to make a hilbert transformer to create $\hat{x}[n]$ (which will be a special FIR filter) and that will introduce a known delay. You must delay all other signals by the same amount to get this relative phase measurement right. The general relation of the discrete Hilbert transform is:

$$\hat{x}[n] = \sum\limits_{m=-\infty}^{\infty} \frac{1 - (-1)^m}{m \, \pi} x[n-m]$$

Of course, you will need to window that infinite sum to a finite sum and delay it to make it causal. Note that all even $m$ terms in the above summation are zero.

\begin{align} \hat{x}[n] &= \sum\limits_{m=-\infty}^{\infty} \frac{1 - (-1)^m}{m \, \pi} x[n-m] \\ &\approx \sum\limits_{m=-2M+1}^{2M-1} w[m] \frac{1 - (-1)^m}{m \, \pi} x[n-m] \\ &= \sum\limits_{m=1}^{2M-1} w[m] \frac{1 - (-1)^m}{m \, \pi} (x[n-m]-x[n+m]) \\ &= \sum\limits_{m=0}^{M-1} w[2m+1]\frac{2}{(2m+1) \pi} (x[n-2m-1]-x[n+2m+1]) \\ &= \sum\limits_{m=0}^{M-1} h[m] \, (x[n-2m-1]-x[n+2m+1]) \\ \end{align}

where $w[n]=w[-n]$ is a decent window (like a Kaiser) of width $4M-1$ and the hilbert coefficients are

$$h[n] \triangleq \frac{2\,w[2n+1]}{(2n+1) \pi}$$

Since this symmetrical, acausal FIR filter looks $2M-1$ samples into the "future", it must be delayed by $2M-1$ samples to be causal and realizable in a real-time context. So you can not realize in real-time a true Hilbert transformer, but you can realize a very close approximation with the corresponding delay of $2M-1$ samples. You must delay all other signals by the same amount to make the following calculations valid. This means you will really be working with $\hat{x}[n-2M+1]$ and you have to explicitly delay $x[n]$ and $y[n]$ by the same amount.

So, while in real-time reality you will be working with $x[n-2M+1]$, $\hat{x}[n-2M+1]$, and $y[n-2M+1]$, for clarity, I am not showing the signals below as so delayed.

Now the measured signal that has the same frequency $\omega$ but different phase is:

$$y[n] = A_y \cos(\omega n + \theta_y)$$

and the number we are looking for is $\theta_y - \theta_x$ .

Consider these two products:

\begin{align} y[n] \cdot x[n] &= A_y A_x \cos(\omega n + \theta_y) \cos(\omega n + \theta_x) \\ &= \frac{A_y A_x}{2} \big( \cos(2\omega n + \theta_y+\theta_x) + \cos(\theta_y - \theta_x) \big) \\ \\ y[n] \cdot \hat{x}[n] &= A_y A_x \cos(\omega n + \theta_y) \sin(\omega n + \theta_x) \\ &= \frac{A_y A_x}{2} \big( \sin(2\omega n + \theta_y+\theta_x) - \sin(\theta_y - \theta_x) \big) \\ \end{align}

Like you have done before, you want to compute the mean of these two products:

\begin{align} \overline{y[n] \, x[n]} & = \frac{A_y A_x}{2} \cos(\theta_y - \theta_x) \\ \\ \overline{y[n] \, \hat{x}[n]} & = -\frac{A_y A_x}{2} \sin(\theta_y - \theta_x) \\ \end{align}

We get the mean by use of any decent low-pass filtering. We don't even need to worry that the DC gain = 1 as you normally do with the mean, since the scaling will come out in the wash. Then, using complex expression:

\begin{align} e^{j(\theta_y - \theta_x)} &= \cos(\theta_y - \theta_x) + j \, \sin(\theta_y - \theta_x) \\ &= \frac{2}{A_y A_x} \left( \overline{y[n] \, x[n]} - j \, \overline{y[n] \, \hat{x}[n]} \right) \\ \end{align}

and

\begin{align} \theta_y - \theta_x &= \arg \bigg\{ \overline{y[n] \, x[n]} - j \,\overline{y[n] \, \hat{x}[n]} \bigg\} \\ &= -\operatorname{atan2}\left(\overline{y[n] \, \hat{x}[n]}, \ \overline{y[n] \, x[n]}\right) \\ \end{align}

\begin{align} \operatorname{atan2}(y,x) &= \begin{cases} \arctan(\frac y x) &\text{if } x > 0, \\ \arctan(\frac y x) + \pi &\text{if } x < 0 \text{ and } y \ge 0, \\ \arctan(\frac y x) - \pi &\text{if } x < 0 \text{ and } y < 0, \\ +\frac{\pi}{2} &\text{if } x = 0 \text{ and } y > 0, \\ -\frac{\pi}{2} &\text{if } x = 0 \text{ and } y < 0, \\ \text{undefined} &\text{if } x = 0 \text{ and } y = 0 \\ \end{cases} \\ \\ &= \begin{cases} \arctan\left(\frac{y}{x}\right) &\text{if } x > 0, \\ \frac{\pi}{2} - \arctan\left(\frac{x}{y}\right) &\text{if } y > 0, \\ -\frac{\pi}{2} - \arctan\left(\frac{x}{y}\right) &\text{if } y < 0, \\ \arctan\left(\frac{y}{x}\right) \pm \pi &\text{if } x < 0, \\ \text{undefined} &\text{if } x = 0 \text{ and } y = 0 \\ \end{cases} \\ \end{align}

• i thought i wrote an answer like this before, but i cannot find it. i know i wrote an answer like this on comp.dsp but i'll never be able to find that. – robert bristow-johnson Jan 25 '17 at 21:34
• Indeed, this will definitely work across the range of $\phi$. – A_A Jan 25 '17 at 22:24
• Sorry for the convetion chosen in my answer. I actually thought about it when I was writing it and I chose $F(t)$ only because for me is the driving Force of my mechanical system (and for me forces have capital letters) and $R(t)$ for the Response signal and I just kept the capital letters. I will keep it as it is now just to not create confusion with all the other comments and answers but I promise that next time I will use the correct convention. Anyway, thank you very much for your answer I will try now to use a second function generator and synchronize them. – Francesco Jan 26 '17 at 14:37
• note also that my answer is in terms of discrete time whereas your question is posed in continuous time. but your solution is likely to be in discrete time. if you are generating your driving function (the $\cos()$ signal), then it should also be possible to generate the quadrature signal ($\sin()$) at the same time. if you are not generating your driving function, but measuring it, just like you're measuring your response, then you need to Hilbert transform that driving function to obtain the quadrature signal that is 90° phase shifted. but you need both to use this method. – robert bristow-johnson Jan 26 '17 at 15:56
• No it's not real time. First I acquire the signal for 1 ms at 10 MHz and then I pass the whole 10E3 samples to FHT. This means that causality is not an issue here. Thanks for the explanations! – Francesco Jan 29 '17 at 17:40

To achieve this, you need to unwrap the phase "signal" $\phi$.

Phase unwrapping works by examining the difference of two successive phase values to determine if a "jump" has occured and correct for it.

In your case, the jump would occur as $\phi$ tends to $180^\circ$ and then wraps around back to $0 ^\circ$ instead of keep "walking" the circle towards $360 ^\circ$ (Or, the other way around the circle).

The way to deal with this would be to get the first derivative of the phase signal and whenever you observe a "jump" larger than $\pi$ you subtract $2\pi$ from instantaneous phase and vice versa, if the jump is smaller than $-\pi$ then add $2\pi$ to the instantaneous phase and keep integrating.

If you want to do this in "real time", then you can establish:

$$\phi_c = \phi + \phi_{offset}$$

Where:

• $\phi_c$ is the corrected phase
• $\phi$ is the instantaneous phase
• $\phi_{offset}$ is the offset ($2\pi, -2\pi$) that is required at a given point in time.

For more information and a worked example, please see this link. Scientific computation platforms such as MATLAB, Scilab, Octave, Python-Scilab and others, usually package this correction in a function that works efficiently over vectors. LabView probably already has a building block that deals with phase unwrapping.

Finally, keep in mind that one of the two signals must have a known phase. That is, either $\phi_F$ or $\phi_R$ must be known or fixed, otherwise, your $\phi$ is relative.

Hope this helps.

• Thanks for the help, I didn't know an algorithm for unwrapping actually exists. However I'm having problems implementing it. The problem is that when $\phi$ is for example $190^{\circ}$ the phase calculation gives me $170^{\circ}$ and not $10^{\circ}$. It doesn't jump back to $0^{\circ}$ but it just goes backward from $180^{\circ}$ towards $0^{\circ}$. How do I do the unwrapping in this case? Thank you very much – Francesco Jan 23 '17 at 18:52
• Can you please edit your main question and provide more details on your specific application and way you acquire your signals? – A_A Jan 24 '17 at 10:29
• I added more information to the main question in the section called EDIT: . Thank you very much – Francesco Jan 24 '17 at 12:37
• the arccos function does not result in a full circle result. he only has 180° to begin with. phase unwrapping does not solve the whole problem. – robert bristow-johnson Jan 25 '17 at 19:40
• @robertbristow-johnson, Thank you, none of the inverse trigonometric functions do though. – A_A Jan 25 '17 at 19:51

Here is a modification to your method that may or may not meet your computational requirements (since I don't know them exactly).

The difficulty in this problem seems to be that there is a phase ambiguity in the $\arccos(\cdot)$ function. To clear this up, we just need to determine which quadrant we are in. One way to accomplish that is to project our signal onto the in phase reference signal and its corresponding quadrature signal (one acting as the cosine for our reference and one acting as the sine).

You have $F(t) = A_F \cos ( \omega t + \phi_F )$ and $R(t) = A_R \cos ( \omega t + \phi_R )$. The modification considers a third signal $F'(t) =A_F \sin \left( \omega t + \phi_F \right)$. If $F(t)$ is generated from a signal generator, then $F'(t)$ could be generated at the same frequency in quadrature with $F(t)$. Otherwise you could delay and/or filter $F(t)$ to obtain $F'(t)$. The rest of the algorithm assumes that you are generating $F'(t)$ the same way you generated $F(t)$. If you need suggestions on how to phase shift the sampled $F(t)$ by $\frac{\pi}{2}$, let me know.

Having $F(t)$ and $F'(t)$ readily available, we can use these steps.

1. Compute the product $X(t) = F(t) R(t)$.
2. Compute the product $Y(t) = F'(t) R(t)$.
3. Compute $\alpha = 2 \mbox{Mean}[ X(t) ]$.
4. Compute $\beta = \mbox{Mean}[ Y(t) ]$.
5. Compute $\phi = -\mbox{sign}(\beta) \arccos \left( \alpha \right)$.

This requires an additional dot product and an additional multiplication by $\pm 1$.

Edit: I tested the above algorithm exhaustively with shifts from -360 to 360 and it seems to work as expected. Let me know if you find problems.

Edit2: This method works perfectly in noise-free environments. I don't know what is the SNR of your signal measurements. I imagine it is fairly high since this sounds like a laboratory problem. I decided to apply a white gaussian noise to my simulations. When I did, I noticed that when noise is sufficiently high, the points near $\phi = \pm \pi$ seem to result in worse estimates than the other points. I worry that this might result in a poor estimate at those points anyway, so it may not resolve your problem. After adding noise, those spots seem to be especially error prone (in terms of noise amplification).

• Thank you very much for the algorithm. However I don't know how to calculate $k$. I need to find where I have a minimum of the phase? But position in which unit? Cycles? – Francesco Jan 23 '17 at 18:54
• It is simply an index. If $|\phi_0|$ is smaller than $|\phi_1|$, then $k=0$. If $|\phi_1|$ is smaller than $|\phi_0|$ then $k=1$. This is what $\mbox{argmin}$ means (find the minimum over the variable--in this case $n$ which can be $0$ or $1$ only--and return the argument). I wrote it this way to save space. – hops Jan 23 '17 at 19:24
• Ok good thank you very much now it is clear. I will implement it in my code and see what I get. Thanks again – Francesco Jan 24 '17 at 9:43
• I'm trying to implement your algorithm, but I'm not sure it works as I would like to. I give you a numerical example: – Francesco Jan 25 '17 at 10:09
• I'm trying to implement your algorithm and I tested it with two simulated signals that I know the phase of. I don't think the algorithm is working as I would like to. I give you a numerical example of when I think it is failing: 1. Phase of simulated $F(t)$ is $110^{\circ}$. 2. Phase of simulated $R(t)$ is $10^{\circ}$. 3. Compute $\phi_0 = 100^{\circ}$. 4. Compute $\phi_1 = 80$. 5. Compute $k = 1$. 6. Then, $\phi = 80 + k \pi = 260^{\circ}$. The phase difference is actually $100^{\circ}$, but I get $260^{\circ}$. Do I need to apply your algorithm only for $\phi_0 \geq 180^{\circ}$ – Francesco Jan 25 '17 at 10:17

Note that if you use an FFT that is of a length that is not an exact integer multiple of the period of your sinusoid, it will distort the phase measurement of your sinusoid, relative to any point in your FFT aperture.

If you can't change the length of your FFT, you can reduce this phase measurement distortion by doing an fftshift of the input vector before the FFT, and/or by using a non-rectangular window (you might try a Blackman-Nutall window when doing phase measurements) if the frequency is high enough relative to the length of your window. Do this before unwrapping phase.

Added: The above also applies to computing just 1 bin of a DFT (or equivalent quadrature mixer) to estimate phase.

• The OP said the FFT operation was too expensive so it is unusable. So performing more operations in addition to the FFT will also be too expensive. – hops Jan 25 '17 at 23:11
• One doesn't need to compute the full DFT or FFT if one knows the frequency. But the same problem applies to 1 DFT result bin, or the equivalent quadrature mixer if there is a vector length mismatch to the period. – hotpaw2 Jan 25 '17 at 23:45


text:
Hyperbolic equilibrium point

In the study of dynamical systems, a hyperbolic equilibrium point or hyperbolic fixed point is a fixed point that does not have any center manifolds. Near a hyperbolic point the orbits of a two-dimensional, non-dissipative system resemble hyperbolas. This fails to hold in general. Strogatz notes that "hyperbolic is an unfortunate name—it sounds like it should mean 'saddle point'—but it has become standard."[1] Several properties hold about a neighborhood of a hyperbolic point, notably[2]

Maps

If ${\displaystyle T\colon \mathbb {R} ^{n}\to \mathbb {R} ^{n}}$ is a C1 map and p is a fixed point then p is said to be a hyperbolic fixed point when the Jacobian matrix ${\displaystyle \operatorname {D} T(p)}$ has no eigenvalues on the unit circle.

One example of a map whose only fixed point is hyperbolic is Arnold's cat map:

${\displaystyle {\begin{bmatrix}x_{n+1}\\y_{n+1}\end{bmatrix}}={\begin{bmatrix}1&1\\1&2\end{bmatrix}}{\begin{bmatrix}x_{n}\\y_{n}\end{bmatrix}}}$

Since the eigenvalues are given by

${\displaystyle \lambda _{1}={\frac {3+{\sqrt {5}}}{2}}>1}$
${\displaystyle \lambda _{2}={\frac {3-{\sqrt {5}}}{2}}<1}$

Flows

Let ${\displaystyle F\colon \mathbb {R} ^{n}\to \mathbb {R} ^{n}}$ be a C1 vector field with a critical point p, i.e., F(p) = 0, and let J denote the Jacobian matrix of F at p. If the matrix J has no eigenvalues with zero real parts then p is called hyperbolic. Hyperbolic fixed points may also be called hyperbolic critical points or elementary critical points.[3]

The Hartman–Grobman theorem states that the orbit structure of a dynamical system in a neighbourhood of a hyperbolic equilibrium point is topologically equivalent to the orbit structure of the linearized dynamical system.

Example

Consider the nonlinear system

{\displaystyle {\begin{aligned}{\frac {dx}{dt}}&=y,\\[5pt]{\frac {dy}{dt}}&=-x-x^{3}-\alpha y,~\alpha \neq 0\end{aligned}}}

(0, 0) is the only equilibrium point. The linearization at the equilibrium is

${\displaystyle J(0,0)=\left[{\begin{array}{rr}0&1\\-1&-\alpha \end{array}}\right].}$

The eigenvalues of this matrix are ${\displaystyle {\frac {-\alpha \pm {\sqrt {\alpha ^{2}-4}}}{2}}}$. For all values of α  0, the eigenvalues have non-zero real part. Thus, this equilibrium point is a hyperbolic equilibrium point. The linearized system will behave similar to the non-linear system near (0, 0). When α = 0, the system has a nonhyperbolic equilibrium at (0, 0).

Comments

In the case of an infinite dimensional system—for example systems involving a time delay—the notion of the "hyperbolic part of the spectrum" refers to the above property.

Notes

1. Strogatz, Steven (2001). Nonlinear Dynamics and Chaos. Westview Press. ISBN 0-7382-0453-6.
2. Ott, Edward (1994). Chaos in Dynamical Systems. Cambridge University Press. ISBN 0-521-43799-7.
3. Abraham, Ralph; Marsden, Jerrold E. (1978). Foundations of Mechanics. Reading Mass.: Benjamin/Cummings. ISBN 0-8053-0102-X.

References

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.


text:
# 10.3: The Reactivity of Multiple Carbon-Carbon Bonds

In the early days of organic chemistry, alkenes were described as "unsaturated" because, in contrast to the "saturated" alkanes, they were found to react readily with substances such as halogens, hydrogen halides, oxidizing agents, and so on. Therefore, the "chemical affinity" of alkenes was regarded as unsatisfied or "unsaturated". (Also see Section 1-1I.)

One reason alkenes and alkynes react more readily than alkanes is because the carbon-carbon bonds of a multiple bond are individually weaker than normal carbon-carbon single bonds. Consider the bond energies involved. According to Table 4-3, the strengths of carbon-carbon single, double, and triple bonds are $$83$$, $$146$$, and $$200 \: \text{kcal}$$, respectively. From these values we can calculate that cleavage of one-half of a carbon-carbon double bond should require $$63 \: \text{kcal}$$ and cleavage of one-third of a carbon-carbon triple bond should require $$54 \: \text{kcal}$$:

As a result, addition reactions to multiple bonds are expected to be about $$20$$-$$30 \: \text{kcal}$$ more exothermic than the corresponding cleavage reactions of carbon-carbon single bonds, as estimated here for reaction with bromine:

The substantial difference in the heats of reaction of ethane, ethene, and ethyne with bromine is reflected in a very important practical consideration in handling ethyne (acetylene), namely its thermodynamic stability relative to solid carbon and hydrogen gas. Unlike ethane, both ethene and ethyne can be shown from bond energies to be unstable with respect to formation of solid carbon and gaseous hydrogen:

Although this does not seem to offer particular problems with ethene, an explosive decomposition of ethyne to carbon and hydrogen may occur if the gas is compressed to $$10$$-$$20 \: \text{kg cm}^{-2}$$. Even liquid ethyne (bp \sim 83^\text{o}\)) must be handled with care. Ethyne is not used commercially under pressure unless it is mixed with an inert gas and handled in rugged equipment. Ethyne burns with pure oxygen to give a very hot flame that is widely used for welding. For this purpose, the gas is dissolved under about $$15 \: \text{kg cm}^{-2}$$ in 2-propanone (acetone, , bp $$56.5^\text{o}$$) and contained in cylinders packed with diatomaceous earth.

Why is ethyne so much less stable than ethene or ethane? First, $$\ce{C-C}$$ bonds are not as strong as $$\ce{C-H}$$ bonds. Therefore a gain in stability usually is to be expected when $$\ce{C-H}$$ bonds are made at the expense of $$\ce{C-C}$$ bonds; ethene and ethane each have more $$\ce{C-H}$$ bonds than ethyne has. Second, ethyne has six electrons held between the two carbons and these electrons experience considerable mutual interelectronic repulsion. This accounts for the fact that the average $$\ce{C-C}$$ bond strength for the triple bond of an alkyne is $$200/3 = 67 \: \text{kcal}$$, compared to $$146/2 = 73 \: \text{kcal}$$ for the double bond of an alkene and $$83 \: \text{kcal}$$ for a normal single bond of an alkane.

## References

John D. Robert and Marjorie C. Caserio (1977) Basic Principles of Organic Chemistry, second edition. W. A. Benjamin, Inc. , Menlo Park, CA. ISBN 0-8053-8329-8. This content is copyrighted under the following conditions, "You are granted permission for individual, educational, research and non-commercial reproduction, distribution, display and performance of this work in any format."

10.3: The Reactivity of Multiple Carbon-Carbon Bonds is shared under a not declared license and was authored, remixed, and/or curated by John D. Roberts and Marjorie C. Caserio.


text:
# How to obtain major mode of current buffer

You can obtain the major mode by using following snippet:

(buffer-local-value 'major-mode (current-buffer))


text:
# Is there a basis for naming these empirical fluid model coefficients in terms of flow regime?

Given any flow restrictive device (e.g. pipe, orifice, screen, etc.) one can measure data as the pressure drop across the device relative to the flow rate through the device. And from this data one can generally closely fit the polynomial

$$\Delta P = K_2Q^2+K_1Q$$

where $\Delta P$ is the pressure drop and $Q$ is the volumetric flow rate

Furthermore there have been people that (I believe) mis-appropriately attribute these factors to concepts involving Reynolds number, calling, for example $K_2$ the 'turbulent' flow factor and $K_1$ the 'laminar' flow factor.

While it is true that $K_1$ dominates the pressure drop at low flow, and $K_2$ dominates at high flow, I don't believe there is any basis to attribute these empirical factors to laminar and turbulent flow characteristics.

Am I maybe missing something? Is there anything in the analysis of fluid dynamics that might support such a naming convention or otherwise refute it?

Suppose geometry of your flow restrictive device is fixed, so that we can take any one geometric dimension of the device (which is relevant to the flow) as your length scale, call it $d$. Assuming fluid properties remain constant, changing flow $Q$ is equivalent to changing average speed $U$ (at any particular cross-section), which we shall take as our velocity scale. Now pressure difference $\Delta p$ required to create this flow depends on fluid properties ($\rho,\mu$), geometric parameters (characterized by $d$), and flow rate (characterized by $U$). Dimensional analysis then gives
$\frac{\Delta p}{\rho U^2}=f(Re)$, where $f$ is some function of Reynolds number.
In a turbulent flow it so happens that pressure drop is determined primarily by processes other than viscous dissipation (mixing of momentum by advection of fluid elements rather than by molecular diffusion), so to a good approximation viscosity is unimportant, in which case there cannot be dependence on $Re$ i.e. $\frac{\Delta p}{\rho U^2} \approx$ constant, which means for a turbulent flow, $\Delta p~\alpha~ U^2~\alpha~Q^2$. I think those who call $K_2$ turbulent flow factor are drawing upon this resemblance. But it is only a resemblence, may be it helps the person remember which factor is which.
If we write out $\Delta p$ as a polynomial series of $Q$, then for small enough $Q$, we may neglect higher powers of $Q$ and we thus have $\Delta p~\alpha~Q$. This is laminar flow regime, and calling $K_1$ laminar flow factor is again only by way of resemblance.


text:
# Abacus

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Jump to: navigation, search

A calculating frame used for arithmetical calculations in ancient Greece, ancient Rome and later, up to the 18th century, in Western Europe. The frame was subdivided into strips, inside which the counters (bones, pebbles, etc.) were moved. The Chinese equivalent of the abacus — swan-pan — was used in the Far East, while the "schetyschety" was employed in Russia.

In nomography an abacus is a special sketch (the so-called net chart nomogram).

#### References

 [a1] W.W. Rouse Ball, "A short account of the history of mathematics" , Dover, reprint (1960) pp. 123–125
How to Cite This Entry:
Abacus. BSE-3 (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Abacus&oldid=18853
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098


text:
# Conditional logit regression -solved using cox regression

I noticed on here a thread which can’t now find about conditional/multinomial logit. Actually, there is no need to implement anything new - this is trivial if cox proportional hazards regression exists and allows for left-censoring. All that needs to be done is to define the ‘start/stop’ times for the various strata as being non-overlapping time intervals but with the same intervals for every member of the same stratum. Then the solution is identical to the conditional logit.

The only difficulty I have is the the required package EventHistory does not seem to be compatible with my Julia 1.2 (I get the dreaded “Unsatisfiable requirements detected” error when I try to use it).
[I have a standard installation of Julia on the latest MacOSX - I guess Julia hasn’t caught up yet]

EventHistory isn’t maintained AFAICT. You can try Survival.jl, which should work on Julia 1.2.

1 Like

Has mlogit and polr models. I plan to add some of the conditional models, but if you can contribute the Cox variant I would be happy to include it.

1 Like

Hi Jose,

Thanks for that.   At this point, all I could do is add an expanded

description because
one of the required packages (EventHistory) for the cox proportional
hazards regression does
not work for Julia on my Mac. However, I have lots of experience
with this trick, and
it has one huge advantage, which is to allow for ties (like in a
race, where two competitors
cannot be separated).

Perhaps if I get access to another computer where Julia runs

properly I may be able to write
and test a script.

Regards,

DavidE

Thank you - that is very useful to know.

I have now loaded Survival.jl, but I can’t seem to find any examples or useful documentation of that package - the very terse documentation for it on the github site is not very helpful for a ‘newbie’ such as myself. Do you know of anywhere I can find an example of someone using this for cox regression in Julia? If I were to have this, I do think I could offer a useful contribution as suggested above.
Thanks in advance for any help.

I agree that the documentation could do with some improvement - and this being open source, you might be the perfect person to do it, once you’ve worked out how to get the results you need!

Fortunately the package has a test suite, so this gives an example of the Cox regression in action:

1 Like

Thank you again for that. It looks like my idea won’t work with this version.
The (apparently unsupported) previous EventHistory had allowed for left-censoring, which was crucial
for applying my idea; It appears that the more recent (and supported) ‘EventTime’ as used here does not allow for that.

There are many reasons why left-censoring is quite essential in a cox regression package,
and my application is a case-in-point. Or perhaps there is another way to run the Cox model using
this package which allows for left-censoring? If not, I would suggest that it be placed high on the
‘to do’ list in order to make this package more widely applicable. If I am missing something here and left-censoring is allowed by the current version then I would love to know how that works. In any case, I am grateful for any info or help on this.

Do you have any resources or good inspiration for an implementation based on mlogit/polr? The infrastructure has been developed for those so I suppose it would only take some flavor syntax,

@formula(response | options ~ id | explanatory variables)

Maybe something like that where the table looks like (for the cmlogit)

outcome option id price income
true    train  1  $50$60,000
false   flight 1  $500$60,000

Then we would only have to adapt the objective function (conditional log-likelihood) and compute the gradient. For ologit, I had to use a trick for the AD to work properly (inspired on the MASS implementation).

I have another (really wacky but effective) approximate (but extremely accurate) solution to this using ordinary logistic regression. Assume the strata are 1,2,…m, and imagine a binary encoding of the first m integers (there will be approximately log(m) base 2 variables required, which is typically not large in relation to n in practice). Assign these to the strata, making factors of them, and then include them with the other input variables when running logistic regression. The resulting model will be nearly identical to the conditional logit, as will the inference, provided the factors are treated as nuisance variables. With slight modification (see below) this circumvents having to write a new routine for conditional logit. As noted above, typically log(m) will be much smaller than n (the sample size) which is important to ensure the numerical stability of the optimisation. [Construction of the binary code is done easily using the base 2 logarithm of each integer and just using the digits]. The above approximation becomes exact if the link function of logit is replaced by the logarithm.
I hope this makes sense and helps. [At least until someone adds left-censoring to the cox regression]

The approach I was considering was Stata’s clogit (page 15 on methods for estimation). Any thoughts on that? The package already has variable absorption so high dimensional features are not an issue. However, I will need to check how code up the gradient of the conditional log-likelihood ("The derivatives of the conditional log-likelihood can also be computed recursively by taking derivatives of the recursive formula f", basically homework). If it is in any of the textbooks like (Stata cites, Greene 2018, chap. 18) it might be easier to find it.

Same for McFadden’s.

If you are talking about the derivatives of the negative log-likelihood, they are quite easy. All you have to do is center the inputs by their probability-weighted averages (as in, estimated probs.), call this matrix x. If the binary output variable is y, then the gradient is merely x’y and the Hessian is given by x’Diag( p )x, where p denote the estimated probabilities (i.e., exp(x b) renormalised by strata).
Does this help? And (by the way) something like R’s clogit would be very useful.

I would have to check if the thresholds do not need to be fitted in that approach. For example, the Hessian in the polr is blockwise where one block is parameters (à la way you described), then there is the parameters cross thresholds, and lastly the last block is the thresholds cross thresholds…
\partial \beta \partial \beta \partial \theta\partial\beta
\partial \theta \partial \beta \partial \theta\partial\theta
Those I could not get manually at all (thanks for AD)… still the gradient also have two components \partial \beta \ \partial \theta.
See the code in for polr.