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# Frequency wavelength multiplexed optoacoustic tomography

## Abstract

Optoacoustics (OA) is overwhelmingly implemented in the Time Domain (TD) to achieve high signal-to-noise ratios by maximizing the excitation light energy transient. Implementations in the Frequency Domain (FD) have been proposed, but suffer from low signal-to-noise ratios and have not offered competitive advantages over time domain methods to reach high dissemination. It is therefore commonly believed that TD is the optimal way to perform optoacoustics. Here we introduce an optoacoustic concept based on pulse train illumination and frequency domain multiplexing and theoretically demonstrate the superior merits of the approach compared to the time domain. Then, using recent advances in laser diode illumination, we launch Frequency Wavelength Multiplexing Optoacoustic Tomography (FWMOT), at multiple wavelengths, and experimentally showcase how FWMOT optimizes the signal-to-noise ratios of spectral measurements over time-domain methods in phantoms and in vivo. We further find that FWMOT offers the fastest multi-spectral operation ever demonstrated in optoacoustics.

## Introduction

Generation of optoacoustic (OA) signals requires illumination of the sample with energy transients (e.g., pulsed or sinusoidal illumination)1. The sample absorbs this time-variant energy and subsequently generates an acoustic wave through thermo-elastic expansion2. Time Domain (TD) OA implementations offer large energy transients by means of nanosecond duration light pulses3,4,5,6, in order to satisfy the thermal and stress confinement limits needed for optoacoustic signal generation7. A nanosecond duration pulse also maximizes the energy transient and optimizes the signal-to-noise ratio (SNR), thus making TD the domain of choice in optoacoustics8,9,10. TD optoacoustic imaging records the time-of-flight of the generated ultrasound waves (US) at multiple locations on the surface of the interrogated object by means of a sensitive ultrasound transducer and, using mathematical inversion, converts these measurements to three-dimensional maps of optical absorption11.

Other imaging modalities such as optical coherence tomography (OCT) or magnetic resonance imaging (MRI) were originally demonstrated in the TD, but have benefited, in terms of imaging speed and SNR, from switching operation to the Frequency Domain (FD)12,13. Frequency Domain (FD) optoacoustics has been also considered as an alternative to TD, by modulating the illumination intensity at a discrete frequency and detecting the generated OA signals at the same frequency14,15,16. Signal detection is achieved with demodulation techniques that retrieve the amplitude and phase of the OA signal, a technology that is simpler and more economic than recording time signals at tens of MHz sampling rates, as is common in TD detection. FD can also enable concurrent illumination at multiple wavelengths, by modulating sources of different color at different frequencies17,18,19. Despite these advantages, intensity-modulated light14,15,16 provides energy transients and corresponding optoacoustic signals that are as low as six orders of magnitude19 weaker than the ultrashort pulses used in TD, drastically reducing the SNR in the FD20,21,22. Moreover, optoacoustic investigations at a single frequency fail to collect depth information or lead to three-dimensional imaging. We have recently shown23 that depth information and three-dimensional image reconstruction requires the generation of signals at multiple discrete frequencies, a requirement that leads to complex emission (modulation) and detection (demodulation) schemes23,24. Therefore, despite the potential advantages over TD17,18,19,23,24, FD has had little impact in the field of optoacoustics. Frequency chirp has also been investigated as a hybrid TD-FD method, by modulating light at a continuously varying frequency17,25, thus encoding time in frequency. Detection is carried out in the TD using time-correlation techniques. Similar to FD methods, however, the use of sine waves limits the achieved SNR, restricting the use of chirp approaches to experimental investigations.

Here, we propose Frequency Wavelength Multiplexed (FWM) operation to categorically improve upon TD performance, while minimizing FD disadvantages. FWM is effectively the reverse implementation of chirp optoacoustics, using a train of discrete pulses, similar to the ones employed in TD optoacoustics, but processing the resulting discrete frequencies that appear in the FD23, a response that is analogous to the frequency comb appearance seen in spectral frequencies. As shown herein, FWM illumination also allows concurrent illumination at multiple wavelengths without increasing the imaging time and yielding an SNR gain that increases by the square root of the number of wavelengths N employed, over TD systems. Following theoretical considerations, we hypothesized that overdriven laser diodes26, which offer a cheaper and more practical alternative to solid-state lasers, could exploit FWM pulse trains and lead to high-quality optoacoustic imaging that could demonstrate benefits over TD implementations. We introduce a frequency wavelength multiplexed optoacoustic tomography (FWMOT) system that utilizes 4 concurrently pulsing overdriven laser diodes, each utilizing a slightly different repetition rate in order to encode different wavelengths. These wavelengths appear then at different discrete frequencies in the frequency domain. We show concurrent multi-wavelength mesoscopic27 imaging of lymphatic and microvascular dynamics in mice at high SNRs, offering the fastest multi-wavelength illumination ever achieved in the field of optoacoustics and confirming spectral performance that improves upon TD implementations.

## Results

### Frequency Comb Optoacoustics using a single excitation wavelength

In conventional TD operation, a square light pulse of duration tp (Fig. 1a) yields a continuous frequency spectrum in the FD, via the Fourier Transform, with the first node at the 1/tp frequency. In FD operation, light modulated by a sine wave in the TD (Fig. 1b) yields a single discrete frequency in the FD. Frequency Wavelength Multiplexed modulation instead considers a train of pulses (Fig. 1c) with a pulse width of tp and a repetition rate frep. The Fourier Transform of this pulse train yields many discrete frequencies with an envelope (Fig. 1c, right) identical to the continuous spectrum of a single pulse with duration tp (Fig. 1a, right). The discrete frequencies of the pulse train are harmonics of the fundamental repetition rate, frep, i.e. integer multiples of frep. A higher repetition rate increases the number of pulses in the TD and reduces the discrete peaks in FD (Fig. 1d). In the FD, a longer pulse width causes the first node (at 1/tp) to appear at lower frequencies (Fig. 1e), reduces the overall energy density at higher frequencies, and lowers the spatial resolution28.

Experimental validation of the FWM scheme was performed by exciting a black varnish layer on a petri dish using only L1 (Fig. 1f, g). Averaging in TD (Fig. 1g, h) increases the SNR by a factor of $$\scriptstyle{\sqrt{{N}_{{{{{\rm{p}}}}}}}}$$, where Np is the number of pulses in the pulse train. In contrast, the proposed FWM method selects the fundamental frequency frep and its harmonics, k * frep, with k being an integer, (Fig. 1i) by performing the following operation (see Supplementary Note 1):

$${S}_{{{{{\rm{a}}}}}}\left(\omega \right)={S}_{{{{{\rm{r}}}}}}\left(\omega \right)\mathop{\sum }\limits_{k=1}^{{{ N}}_{{{{{\rm{h}}}}}}}\delta \left(\omega -{k\;\omega }_{0}\right),$$
(1)

where Sa (ω) is the Fourier Transform of the averaged signal, Sr (ω) the Fourier Transform of the recorded signal (Fig. 1f), Nh the number of the harmonics in the detected ultrasound transducer bandwidth and ω0 = 2π/T. The OA signal is generated at the exact same frequencies that the excitation LD pulse train contains. Therefore, operation (1) selects only the harmonics of frep and filters out frequencies that do not contain signal to increase the SNR by the same factor $$\scriptstyle{\sqrt{{N}_{{{{{\rm{p}}}}}}}}$$ as in the TD (Fig. 1j). The signal in Fig. 1j is the Fourier Transform of the signal in Fig. 1h, with the two signals matching perfectly (Fig. 1k). This analysis confirms that FWM illumination results in a practical generation of multiple discrete frequencies, required for accurate FD operation, offering an SNR that is equivalent to the TD when a single wavelength is used. FWM analysis is, therefore, the Fourier space equivalent to normal time averaging. Next, we show however that FWM offers advantages over TD when multiple wavelengths are employed.

### Frequency Comb Optoacoustics using multiple excitation wavelengths

To demonstrate the FWM advantage over the TD (Fig. 2) we plotted the single-wavelength excitation pattern in the TD and its power spectrum in Fourier space (Fig. 2a) to serve as reference for the analysis that follows. The pulse train shown has a period T that corresponds to a repetition rate frep = 1/T, a total number of pulses Np, a pulse width tp and an acquisition time tacq = Np × T. The period T defines the maximum depth-of-view DoV = vs × T that can be achieved for the pulse train selected, where vs is the speed of sound.

TD wavelength multiplexing is performed using wavelength interleaving, or time-sharing. However, when increasing the number of wavelengths in the TD, at least one of the following three parameters has to be compromised: the DoV, the number of pulses in the pulse train and therefore the SNR for each wavelength, or the total acquisition time. Figure 2b shows how the DoV is reduced when using four wavelengths at a given total acquisition time. The different wavelengths excite the tissue using the same repetition rate frep but with a time shift tsh between each wavelength (Fig. 2a), given by tsh = T/N, where N is the number of wavelengths. The result is a reduction of the time between subsequent pulses, limiting the DoV available to each wavelength by a factor N. Alternatively, it is possible to retain the original DoV by dropping the repetition rate for each wavelength, j, to frep,j = frep/N and the number of pulses per wavelength to Np/N (Fig. 2c); however, this approach results in an SNR reduction by a factor of $$\sqrt{N}.$$ A third alternative retains the original DoV and SNR while prolonging the acquisition time by a factor N (Fig. 2d).

In contrast to TD, FWMOT employs a small frequency shift δf for the repetition rate of each wavelength (Fig. 2e; time), whereby δf << frep. Each wavelength has a different repetition rate, which results in a slightly different effective DoV; however, since δf << frep, this difference is insignificant. The power spectrum of the FWMOT excitation pattern (Fig. 2e; right) shows the appearance of harmonics from the fundamental repetition rate for each wavelength. The repetition rate of the first laser, frep,1, can be selected as the reference repetition rate, with the repetition rate of the remaining lasers given by frep,j = frep,1 + (j − 1) * δf. FWMOT thus recovers the OA signals by evaluating Eq. (1) with the corresponding ω0 = 2πfrep,j for each wavelength. Consequently, using frequency separation, FWMOT can multiplex different wavelengths without compromising the DoV, SNR or acquisition time. With the signal processed in the FD, the frequency resolution is defined by df = 1/tacq, where tacq = Np/frep,1, meaning that frequencies that differ by less than df cannot be resolved. In order to recover the OA signal from all wavelengths, all harmonics from all lasers that lie in the Ultrasound Transducer (UST) detection bandwidth should therefore be spaced at a distance exceeding the frequency resolution df, imposing a minimum number of pulses Np,min that depends on the UST bandwidth limits (flow and fhigh), N and frep,1 (see Supplementary Note 2). The parameters N, flow, fhigh, frep,1 and the chosen number of pulses, Np, define the range of small frequency shifts (δf), between δfmin and δfmax, required to recover the signal of each wavelength without losses. A δf value larger than δfmin ensures that harmonics of laser j and j + 1, which lie on the lower end of the UST bandwidth, are well resolved, while a δf value smaller than δfmax ensures that harmonics of laser 1 and N, which lie on the upper end of the UST bandwidth, are well resolved.

### FWMOT outperforms TDOA in SNR, DoV or imaging speed

To experimentally demonstrate the advantages of FWM operation in multi-wavelength excitation compared to TD optoacoustics, we employed FWMOT with all four wavelengths. The resulting optoacoustic signal at wavelength 1, frep,1 = 200 kHz and Np = 200 is presented in Fig. 2f and attained tacq = 1 ms and DoV = 7.5 mm. Electromagnetic interference from the laser diode (LD) driving circuit is indicated with α. The primary OA signal generated from black varnish on a petri dish is indicated with β, whereby reflections from the UST glass lens that arrive later in time are indicated with γ. The “Noise Level” inset shows a separate measurement of the noise level that yielded a standard deviation of 90.6 μV. The intensity of the OA signal from LD1 was measured to be 12 mV, resulting in an SNR of 21.2 dB.

Figure 2g presents the averaged OA signal in the TD, when the DoV is compromised. Each LD has frep,j = 200 KHz, Np = 200 and a time shift between each other that results in DoV = 1.875 mm and tacq = 1 ms. The OA signal resulting from the simultaneous excitation using the pattern in Fig. 2b is presented as a green line and is the sum of the individual OA signals obtained when each wavelength was pulsed separately (Fig. 2g). We could easily detect the laser trigger interference for all wavelengths (α). The OA signals from wavelength 1 and 2 (β) are located very closely to the laser interference (α) of wavelengths 3 and 4 respectively, vastly reducing their SNR (5.7 dB for wavelength 1). However, the OA signal of wavelength 3 is completely masked by the interference of wavelength 1. The reflections of the OA signal from wavelength 1, 2 and 3 (γ) are still visible. Therefore, electromagnetic interference and OA reflections further limit the DoV and SNR achieved in multiple wavelength TD optoacoustics.

Likewise, SNR limits are imposed (Fig. 2h) in response to an excitation pattern (Fig. 2c) that uses frep,j = 50 kHz, Np = 50 and DoV = 7.5 mm for each wavelength with tacq = 1 ms. For all wavelengths, we observed that the laser interference (α), the OA signal (β), and the reflections (γ) are all visible but with lower SNR. In this case, the noise standard deviation (presented in the Figure insets) was 161.6 μV, which is higher than the 90.6 μV measured in the pulse train with 200 pulses to average (Fig. 2f), while the signal intensity remained 12 mV resulting in an SNR of 18.8 dB for wavelength 1. Finally, with frep,j = 50 kHz, Np = 200, and DoV = 7.5 mm for each laser, the OA signal of all 4 lasers can be recorded without SNR losses (N = 90.6 μV and S = 12 mV with 21.2 dB for wavelength 1) but with a longer acquisition time, tacq = 4ms, that is presented in Fig. 2i.

Conversely, FWMOT operation for four wavelengths uses frep,1 = 200 kHz, δf = 125 Hz and Np = 200 and recovers each signal without cross-talk between wavelengths (Fig. 2j), experimentally confirming theoretical predictions. Supplementary Figs. 4 and 5 present the Laser Diode excitation time series, the corresponding OA signal time series and the power spectra of both, in order to further showcase FWMOT’s operation and verify the method’s ability to recover each signal without cross-talk (see Supplementary Note 3). FWMOT is able to provide high SNR for concurrent excitation with multiple wavelengths, without extending the acquisition time (1 ms) and achieving the same DoV (7.5 mm) and SNR (here again N = 90.6 μV and S = 12 mV, resulting in an SNR of 21.2 dB for wavelength 1).

We would expect an SNR increase of 3 dB upon increasing the numbers of averaged measurements from 50 to 200. However, we measured an SNR increase of 2.3 dB for both TD averaging and FWM. This discrepancy can be attributed to the presence of systematic electromagnetic noise in the system, which is also averaged and reduces the SNR by 0.7 dB. However, this effect is a general feature of the OA system and not a disadvantage of the FWM algorithm.

We also compared the SNR obtained in conventional FD optoacoustics to FWMOT by employing the same black varnish phantom and single-wavelength illumination at 445 nm (see Supplementary Fig. 1). FD optoacoustics employed a sine wave of 20 MHz frequency with adjusted mean power to equal the mean power output of the FWMOT pulsed pattern used for 6.8 ns pulses at 200 KHz. FWMOT demonstrated an SNR that was 20.8 dB higher compared to FD excitation.

### FWMOT multi-wavelength imaging of tissues and tissue dynamics in vivo

While the theoretical merits of FWM optoacoustic operation were demonstrated with phantom measurements, a next critical step was to examine whether FWM could offer realistic implementations. For this reason, we aimed to investigate whether the theoretical advantages could lead to operational characteristics (SNR, acquisition speed) that would render FWMOT appropriate for in vivo applications. A particular unknown parameter in this interrogation was the FWMOT performance achieved with multiple wavelengths using laser diodes, as it would be impractical to implement FWMOT with multiple solid-state lasers due to cost and size. To examine the merits of using low-cost technology, we investigated the performance of multiple laser diodes to image vasculature and lymphatics in vivo, using the mouse ear as a model. This imaging target was selected because it is a typical tissue on which TD optoacoustic implementations have been conventionally demonstrated.

First, we assessed whether FWMOT could produce high-quality images from biological specimens. We employed FWM illumination at 445 nm and 465 nm and resolved oxy- and deoxy- hemoglobin (Fig. 3a, b) based on their spectral difference, with de-oxygenated hemoglobin absorbing higher at 445 nm, and vice-versa. The data were collected on a grid and were first averaged point-by-point using the FWM algorithm before being fed into the reconstruction algorithm (see methods). Superposition of Fig. 3a, b revealed a color-coded composite image (Fig. 3c) of the relative vascular oxygenation, with red color corresponding to higher oxygenation levels. We further confirmed that FWMOT can produce depth-resolved images (see Supplementary Fig. 2) without cross-talk between the wavelengths, offering first evidence that FWMOT can acquire images from biological tissues based on LDs.

We next aimed to identify whether FWMOT could visualize multiple moieties in tissue without compromising operational characteristics as in the TD. We introduced exogenous contrast by intradermal injection of Evan’s Blue and Indocyanine Green (ICG) and applied 4-wavelength FWMOT to simultaneously resolve arteries and veins (Fig. 3d, e) and lymphatic vessels revealed by contrast enhancement (Fig. 3f, g). Figure 3h shows the corresponding bright-field image of the mouse ear. The composite image of four wavelengths (Fig. 3i) enabled visualization and co-localization of the vascular and lymphatic vessels. Notably, the use of an ultra-wideband transducer enabled to resolve the fine structures represented by the vessels as well as the large absorbing areas that were formed around the injection sites. FWMOT can therefore be effectively used to perform OA imaging with multiple wavelengths simultaneously in vivo with an acquisition of ~30 min, while the same multispectral implementation in TD requires ~2 h, rendering such a measurement impractical.

The FWM acquisition acceleration over TD demonstrated in Fig. 3 also points to an FWMOT use for detecting rapid changes simultaneously using multiple wavelengths. We therefore applied FWMOT to monitor oxygenation fluctuations in the mouse ear during an oxygen stress test in vivo. Figure 4a shows the composite OA image from 2 blue wavelengths revealing the oxygenated and de-oxygenated vessels. We selected a 2 mm line (Fig. 4a; blue box) to acquire signals repeatedly (continuous FWM operation) at a rate of ~4 Hz. A B-scan (indicated by the letter (i) on Fig. 4a) revealed an artery (red) and a vein (green) as a function of depth. The oxygen challenge was provided by alternating the composition of the breathing gas from 0.8 liters per minute (lpm) of 100% O2 (Fig. 4b; “O2”) to 0.6 lpm of 20% O2 plus 0.2 lpm CO2 (Fig. 4b; “Air”). Figure 4b plots the ratio of the wavelength 2 signal (S2) over the wavelength 1 signal (S1) over time. The S2/S1 ratio is indicative of the relative changes of oxygenation in the vessels over time. As expected, higher oxygenation was observed in the artery than the vein throughout the experiment. Oxygenation was reduced during the supply of air and increased when 100% O2 was provided. The oxygenation levels in the artery increased faster in response to the change from Air to O2 supply compared to the vein, revealing the expected dynamics of oxygen supply to tissues.

The oxygen extraction rate (OER in Fig. 4c), an indication of the oxygen uptake by cells, was calculated as OER = (OcaOcv)/Oca, whereby Oca and Ocv is the oxygen saturation in the central artery and vein, respectively29. We observed that OER slightly dropped shortly after the Air supply period, suggesting a delayed response in cell oxygenation. When the oxygen supply was increased during the second and third O2 supply periods, an increased OER was observed followed by a return to normal levels, suggesting that cells consumed more oxygen as it became available. The oxygen stress test experiment confirms FWMOT as a method that can be employed in studying tissue dynamics with high localization ability, confirmed by intravenous injection of ICG and Evan’s Blue. Using the same 2 mm observation field, FWMOT recorded contrast agent dynamics at 638 and 808 nm (Fig. 4d). We observed a similar post-injection pattern for both dyes, revealing two distinctive peaks before settling to a baseline value, indicative of the circulation dynamics of the dyes in the vascular system. We further resolved the delayed appearance of the agents in the vein for both dyes, whereby the observed signal increased at a lower rate, compared to the artery, a pattern attributed to dye diffusion in the tissue capillary network.

## Discussion

We introduce an optoacoustic method based on frequency multiplexing, which challenges the prevailing notion that TD implementations offer the best optoacoustic performance. By combining the advantages of TD excitation, i.e. pulsed excitation, with FD analysis, FWMOT offers superior performance to TD or FD at multi-wavelength excitation. Increasing the number of wavelengths (N) in TD compromises the depth-of-view, SNR, or the total acquisition time. FWMOT can provide simultaneous illumination at multiple wavelengths without sacrificing any of these parameters, enabling either a higher SNR by a factor of $$\sqrt{N}$$ per wavelength, shorter acquisition time by a factor of N, or N times more depth-of-view per wavelength compared to TDOA. Moreover, the use of pulse trains leads to practical FWMOT implementations by providing the necessary discrete frequencies in the frequency domain, while avoiding generation of FD signals by direct operation in the frequency domain using elaborate multi-frequency modulation systems.

Another critical aspect herein was not only to show the theoretical superiority of the method, but also to demonstrate that it can lead to practical implementations. An enabling technology that allowed such performance was the use of overdriven continuous wave laser diodes (CW-LD)26 which are inherently cost-effective, portable, and compact, leading to systems with the potential of high dissemination. LDs are ideally suited for FWMOT and can be used to replace solid-state lasers. Besides their small form factor and availability at multiple wavelengths, LDs can be pulsed at very high pulse repetition rates, matching the uniquely optimal FWMOT ability to multiplex and average signals, increasing the SNR. New and more powerful LDs entering the market will contribute to further improvements of FWMOT performance, offering compact and low-cost systems with high dissemination potential for various applications9,27.

Imaging at four wavelengths differentiated blood vasculature from lymphatic vessels in vivo, reducing the acquisition time from 2 h required in TD to ~30 min. FWMOT also achieved a fourfold higher acquisition rate compared to TD when performing B-Scans over the main artery and vein, enabling the monitoring of relative oxygenation changes and calculation of oxygen extraction rate. Therefore, FWMOT is particularly suited for dynamic measurements or for experiments that require anesthesia, improving also the associated throughput rate over TD implementations.

There is also a limit to the maximum number of wavelengths used in FWMOT that depends on the UST detection bandwidth, the acquisition time, or the number of averages in the pulse trains and the repetition rate of L1 used in each case. When frep,1 = 200 kHz, Np = 100, and the UST bandwidth between 22 and 78 MHz, the maximum number of wavelengths is 28 for FWMOT compared to just 5 for TD implementations with the same operating parameters and a DoV of 1.5 mm per wavelength (see Supplementary Note 4). The more wavelengths used in FWMOT, the greater the SNR increase per wavelength compared to TDOA. The use of many simultaneous wavelengths would be particularly appealing for improving the molecular detection specificity of OA spectroscopy.

In summary, FWMOT enables fast, high SNR imaging using multiple wavelengths simultaneously without compromising the DoV and can offer a valuable tool for studying dynamic molecular processes, revolutionizing how multispectral OA imaging will be performed in the future.

## Methods

### 4 Laser diode raster scanning optoacoustic tomography system for FWMOT

Figure 5 presents the multispectral raster scanning optoacoustic mesoscopy (RSOM) system developed to test the advantages of FWMOT. Matlab (Matlab 2016b, Mathworks, USA) was installed on a PC, controlling the system (Fig. 5a). The PC controls the dual-channel stages driver (C-867.260, Physik Instrumente, Germany) that drives the dual xy stages placed on the Scanning Head (Fig. 5c) and is synchronized with two Arbitrary Waveform Generators (AWG, 33522B, Keysight, USA). The AWGs trigger the laser diode drivers and provide a synchronization pulse to the Data Acquisition Card (DAQ, 12-bit, 200MS/s, Razor Express 14x2 Compuscope, Dynamic Signals LLC, USA) for synchronous acquisition of the signal. The light output of the Illumination system is directed into the Scanning Head, with a small portion entering a photodiode (DET10A2/M, Thorlabs, USA). The signal from the photodiode is analog filtered (BLP-90+, Minicircuits, USA) and recorded to monitor the pulse to pulse energy fluctuations and time jitter, corrected for both during signal post-processing. The OA signal from the UST is amplified with a 60dB gain amplifier (Miteq AU-1291-R, Miteq, USA) and analog filtered (BLP-90+ and ZFHP-1R2-S+, Minicircuits, USA) before being digitized by the DAQ to avoid aliasing.

The Illumination system (Fig. 5b) consists of four CW laser diodes that were overdriven with four high-current, short-pulse laser diode drivers developed previously26. In short, the peak current of the LDs is briefly (for nanoseconds) increased to >40-fold their CW absolute maximum, which allows the LDs to provide up to 27-fold higher peak power than the manufacturer specified absolute maximum limit. The laser diodes used in this work are the LDM-445-6000 (LaserTack, Germany) emitting at 445 nm, the LDM-465-3500 (LaserTack, Germany) emitting at 465 nm, the HL63283HG (Ushio, Japan) emitting at 638 nm and the K808D02FN (BWT, China) emitting at 808 nm, named laser 1, laser 2, laser 3 and laser 4 respectively. Each laser diode is focused in a multimode fiber. In order to position each laser diode in a manual X–Y stage (CXY1, Thorlabs, USA), a collimating lens (C340TMD, Thorlabs, USA) is placed in front of it on a manual z-stage (SM1Z, Thorlabs, USA), followed by a focusing lens (C560TME, Thorlabs, USA) that is kept stable and the fiber on a x–y stage (CXY1, Thorlabs, USA). The fiber with a 200um core diameter and 0.22NA was one of the 4 inputs of 4x4 fiber power combiner. The four outputs of the fiber combiner (MPC-4-M21-M41-P23, Lasfiberio, China) contain ~25% of the input power of each input fiber and are also multimode fibers with a 200 µm fiber core and 0.22NA. One of the outputs is connected to a custom made 95-5% splitter (LTL 500-93310-95-1, LaserComponents Germany GmBH, Germany) and the 5% fiber was connected to the photodiode. The 3 outputs of the power combiner and the 95% fiber of the splitter were terminated with 1.25 mm ferrules (SFLC230, Thorlabs, USA) and directed to the Scanning Head.

The Scanning Head (Fig. 5c) consists of the x–y stage (U-723 XY, Physik Instrumente, Germany), the 3D printed holder, the ultrasound transducer (UST, HFM23, Sonaxis, France) with a 50 MHz central frequency and 112% relative bandwidth, 3 mm focal length and 0.5 NA, and the four output fibers arranged in a circular pattern around the UST. The output of the four fibers is designed to cross at the focal spot of the UST to achieve a maximum energy density on the sample.

Scanning and recording are performed in a sweeping-like motion using the stages driver as the master in the system. The x-stage moves in a straight line with constant velocity. When the x-stage has traveled a specific distance, equal to the step size, the stages driver sends a signal to the AWGs to trigger the laser diodes. Each laser diode is triggered using a pulse train with repetition rate frep,1frep,4 for each laser diode and Np1Np4 number of pulses at each point (A-Scan) in the B-Scan. After the stage has traveled the desired distance, the B-Scan is completed and the stages stop moving. The y stage is moved to the next y-position and the x-stage can now perform the next B-Scan in the opposite direction. For the FWMOT excitation we used frep,1 = 200,000 Hz, frep,2 = 200,125 Hz, frep,3 = 200,250 Hz and frep,4 = 200,375 Hz and Np1 = 100 and Np2 = Np3 = Np4 = 101 pulses for lasers 1, 2, 3, and 4, respectively. For a detailed derivation of these values, see Supplementary Note 2.

We recorded the emission spectra of the 4 laser diodes used in the multispectral laser diode RSOM with a spectrometer (USB4000, OceanOptics, UK) and the peak emission wavelengths were 444.3, 460.1, 636.8, 804.9 nm with a variance of 1.6, 1.7, 1.9, 2.2 nm respectively and an R2 confidence level higher than 0.96 for all cases.

Additionally, the energy per pulse on samples was measured with a stabilized thermal power meter (PM160T, Thorlabs, USA) and calculated as 189, 137, 142, 153 nJ per pulse for lasers 1, 2, 3, 4, respectively. The pulse width was estimated as 6.7, 6.7, 10.2, 10.2 ns full-width-at-half-maximum (FWHM) for each laser respectively. Using a USB CCD camera (daA1920-30 µm; Basler AG, Germany) the illumination spot on the surface of the sample was measured to be a circle with a diameter of ~1 mm. Table 1 presents a summary of the LD pulsing and emission characteristics for FWMOT operation as presented above.

To compare the FD OA to FWMOT excitation we used a fiber-coupled 450-nm laser diode (FBLD-450-0.8W-FC105-BTF; Frankfurt Laser Company, Germany) connected to an analog laser driver (BFS-VRM 03 HP; Picolas, Germany). The output fiber of the laser was pumped into one of the input fibers of the 4 × 4 fiber combiner, so that the illumination of the sample is identical for the FD excitation and the FWMOT excitation system.

To calculate the Signal-to-Noise ratio (SNR) we used the following formula, 10 * log10(S / N), where S is the signal intensity and N the standard deviation of the noise floor.

### Signal processing, reconstruction algorithm, and system resolution

Image acquisition occurs in a large Field of View (10 × 10 mm2) and with a scanning step size of 10 µm in order to be much lower than the lateral resolution of the system. Time-resolved signals detected at each scanning position correspond to the integration of acoustic spherical waves originating from the illuminated optical absorbers within the detection angle of the transducer. Therefore, unprocessed images obtained directly from the system are heavily blurred, and require further processing to obtain high-contrast and high-resolution images. To do so, a back-projection algorithm is implemented in the Fourier domain30,31 to recover an (acoustically) diffraction-limited image. The resulting image is then corrected by the system impulse response32 and further processed with a vesselness filter for display purposes33. The raw data of a single B-Scan do not allow for 3D image reconstruction. Therefore, a simplified version of the back-projection algorithm was developed, which operates in two dimensions and approximates the transducers sensitivity field as a conic section along the B-Scan direction only.

To calculate the system’s spatial resolution we imaged a resolution target (a Siemen’s star) using all 4 wavelengths. The results are presented in Supplementary Fig. 3, which shows the reconstructed OA images of the resolution target at all four wavelengths, as well as the composite image in polar coordinates. The spatial resolution is given by the smallest radius at which all the lines can be well resolved (Supplementary Fig. 3h), which is 38 µm for all wavelengths.

### Laser diode emission spectra, hemoglobin and dye absorption spectra

Hemoglobin has a broad absorption spectrum over the visible and near-infrared range with a higher absorption at the lower wavelengths of the spectrum. The absorption of de-oxygenated hemoglobin is higher than the absorption of oxygenated hemoglobin at 444 nm. The absorption of oxygenated hemoglobin is higher than the absorption of de-oxygenated hemoglobin at 460 nm. The total absorption of hemoglobin at wavelengths 1 and 2 is much higher than that at wavelengths 3 and 4. Due to the low energy output of the laser diodes, this is enough to assume that we only detected OA signals from hemoglobin at wavelengths 1 and 2. This has been confirmed from the in vivo mouse ear experiments. Moreover, we can estimate the relative changes of oxygen saturation34 by calculating the ratio of the signal intensity of wavelength 2 over that of wavelength 1, S2/S1, with a higher ratio indicating a higher oxygen saturation.

To induce contrast and increase the SNR at wavelengths 3 and 4 we used two dyes, Evan’s Blue (Sigma-Aldrich, Germany) and ICG (VERDYE, Germany). Evan’s Blue has a peak absorption at 640 nm and a minimum at 740 nm35 making it the appropriate dye to enhance the contrast for wavelength 3, since this is the only wavelength where we can detect OA signals. ICG in blood plasma has a peak absorption at around 810 nm and a low absorption at 637 nm36, making it appropriate to induce OA contrast at wavelength 4 with minimal contrast at wavelength 3 and no contrast at the other wavelengths, confirmed by the mouse ear experiments.

### Maximum permissible exposure (MPE) limits compliance and presence of non-linear effects

Using the above-mentioned values (Table 1) for each laser diode and a scanning speed of 10 mm/s for a scanning step size of 10 μm we can calculate the sample exposure. The total exposure of the sample for simultaneous illumination with all four wavelengths is calculated to be 19.8 µJ/cm2 per pulse and 3.96 W/cm2 mean exposure, well below the MPE limits of 20 mJ/cm2 and 18 W/cm2, imposed by the American National Standards Institute37. It has been demonstrated that non-linear effects in OA are present at high energy densities above ~7 mJ/cm238. The total sample exposure herein was less than 20 µJ/cm2, even in the case of simultaneous illumination with all wavelengths. Therefore, the deposited energy in this study was much lower than the threshold for non-linear effects.

### Mouse handling and imaging protocol

For our experiments, we employed two 5- to 6-week-old female Athymic nude- Foxn1nu mice (Envigo, Germany). During all measurements, the mice were anesthetized by 1.6% Isoflurane (cp-Pharma, Germany) with 0.8 lpm carrier gas flow and body temperature was maintained with an infrared heat lamp and heating plate.

The first mouse was used for the experiments presented in Fig. 3. The lymphatic ear vessels were highlighted by intradermal administration of 5 µl ICG (5 mg/ml) and 5 µl Evan’s blue (1%) into the ear tip of the mouse. ICG was administered ~30 min and Evan’s blue ~10 min before starting imaging to ensure lymphatic drainage of the dyes.

A second mouse was used for the experiments demonstrated in Fig. 4. For the oxygen stress experiment, we supplied different isoflurane carrier gas combinations or breathing conditions through a nose mask. The mouse was breathing alternatively 0.8 lpm of 100% oxygen (O2) and a combination of 0.6 lpm medical air (20% Oxygen) plus 0.2 lpm carbon dioxide CO2 (Air). For the dye diffusion experiment we first acquired the background data and after 3 min, intravenously injected 100 µl of 1% Evan’s Blue solution, and after another 3 min, 100 µl of the 5 mg/ml ICG solution. Both mice were sacrificed immediately after imaging by cervical dislocation.

All mouse experiments were performed according to the committee on Animal Health and Care of Upper Bavaria, Germany (Az. ROB-55.2-2532.Vet_02-14-88 and Az. ROB-55.2-2532.Vet_02-18-120). The mice were maintained in an individual ventilated cage system (Tecniplast, Germany) at 22° ambient temperature, ~50% relative humidity, and with a regular 12 h day/night cycle in our specific-pathogen-free (SPF) mouse facility at the Center for Translational Cancer Research of the Technical University of Munich (TranslaTUM).

### Human skin experiment

One of the authors volunteered for this study to have their hand imaged. After consultation with the TUM Ethics Commission, no formal ethics approval was necessary. Informed consent from the participant was obtained and archived. The participant is a 30-year-old white male. Image acquisition occurred in a Field of View of 5 × 5 mm2 and with a scanning step size of 10 µm. The reconstructed OA images are presented in Supplementary Fig. 2.

### Reporting summary

Further information on research design is available in the Nature Research Reporting Summary linked to this article.

## Data availability

The raw optoacoustic signal data for validating the frequency wavelength multiplexed optoacoustic tomography algorithm, the raw imaging data of mice and humans, the raw B-Scan data of the Oxygen stress test and ICG and Evan’s Blue injection experiments that were generated in this study have been deposited in the Zenodo database under accession code: https://doi.org/10.5281/zenodo.6770729.

## Code availability

The authors declare that for data collection the commercially available software from GaGe (Compuscope Driver Version 5.04.34, Dynamic Signals LLC, USA) and Matlab 2016b (Matlab, Mathworks, USA) was used. Data analysis was conducted in Matlab using its built-in functions. The frequency wavelength multiplexed optoacoustic tomography algorithm has been patented and is available upon discretion from the corresponding author.

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2. Xu, M. & Wang, L. V. Photoacoustic imaging in biomedicine. Rev. Sci. Instrum. 77, 041101 (2006).

3. Diebold, G. J., Sun, T. & Khan, M. I. Photoacoustic monopole radiation in one, two, and three dimensions. Phys. Rev. Lett. 67, 3384–3387 (1991).

4. Calasso, I. G., Craig, W. & Diebold, G. J. Photoacoustic point source. Phys. Rev. Lett. 86, 3550–3553 (2001).

5. Taruttis, A. & Ntziachristos, V. Advances in real-time multispectral optoacoustic imaging and its applications. Nat. Photonics 9, 219–227 (2015).

6. Brecht, H.-P. et al. Whole-body three-dimensional optoacoustic tomography system for small animals. J. Biomed. Opt. 14, 064007 (2009).

7. Wang, L. V. Tutorial on photoacoustic microscopy and computed tomography. IEEE J. Sel. Top. Quantum Electron. 14, 171–179 (2008).

8. Ntziachristos, V. & Razansky, D. Molecular imaging by means of Multispectral Optoacoustic Tomography (MSOT). Chem. Rev. 110, 2783–2794 (2010).

9. Aguirre, J. et al. Precision assessment of label-free psoriasis biomarkers with ultra-broadband optoacoustic mesoscopy. Nat. Biomed. Eng. 1, 1–8 (2017).

10. Omar, M., Schwarz, M., Soliman, D., Symvoulidis, P. & Ntziachristos, V. Pushing the optical imaging limits of cancer with multi-frequency-band raster-scan optoacoustic mesoscopy (RSOM). Neoplasia 17, 208–214 (2015).

11. Rosenthal, A., Ntziachristos, V. & Razansky, D. Acoustic inversion in optoacoustic tomography: a review. Curr. Med. Imaging Rev. 9, 318–336 (2013).

12. de Boer, J. F., Leitgeb, R. & Wojtkowski, M. Twenty-five years of optical coherence tomography: the paradigm shift in sensitivity and speed provided by Fourier domain OCT [Invited]. Biomed. Opt. Express 8, 3248 (2017).

13. Kumar Jhamb, T., Rejathalal, V. & Govindan, V. K. A review on image reconstruction through MRI k-space data. Int. J. Image, Graph. Signal Process. 7, 42–59 (2015).

14. Kellnberger, S., Deliolanis, N. C., Queirós, D., Sergiadis, G. & Ntziachristos, V. In vivo frequency domain optoacoustic tomography. Opt. Lett. 37, 3423–3425 (2012).

15. Mohajerani, P., Kellnberger, S. & Ntziachristos, V. Frequency domain optoacoustic tomography using amplitude and phase. Photoacoustics 2, 111–118 (2014).

16. Telenkov, S., Mandelis, A., Lashkari, B. & Forcht, M. Frequency-domain photothermoacoustics: Alternative imaging modality of biological tissues. J. Appl. Phys. 105, 1–8 (2009).

17. Mandelis, A. & Guo, X. Wavelength-modulated differential photothermal radiometry: theory and experimental applications to glucose detection in water. Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 84, 1–14 (2011).

18. Lashkari, B., Sean Choi, S. S., Khosroshahi, M. E., Dovlo, E. & Mandelis, A. Simultaneous dual-wavelength photoacoustic radar imaging using waveform engineering with mismatched frequency modulated excitation. Opt. Lett. 40, 1145 (2015).

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32. Omar, M. et al. Optical imaging of post-embryonic zebrafish using multi orientation raster scan optoacoustic mesoscopy. Light Sci. Appl. 6, e16186–e16186 (2017).

33. Longo, A. et al. Assessment of hessian-based Frangi vesselness filter in optoacoustic imaging. Photoacoustics 20, 100200 (2020).

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35. Radu, M. & Chernoff, J. An in vivo assay to test blood vessel permeability. J. Vis. Exp. 2–5 https://doi.org/10.3791/50062 (2013).

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37. Delori, F. C., Webb, R. H. & Sliney, D. H. Maximum permissible exposures for ocular safety (ANSI 2000), with emphasis on ophthalmic devices. J. Opt. Soc. Am. A 24, 1250 (2007).

38. Malekzadeh-Najafabadi, J., Prakash, J. & Ntziachristos, V. Nonlinear optoacoustic readings from diffusive media at near-infrared wavelengths. J. Biophotonics 11, e201600310 (2018).

## Acknowledgements

The research leading to these results has received funding by the Bundesministerium für Bildung und Forschung (BMBF), Bonn, Germany (Project Sense4Life, 13N13855) (V.N.), from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program under grant agreement No 694968 (PREMSOT) (V.N.) and from the European Union’s Horizon 2020 research and innovation program under grant agreement No 732720 (ESOTRAC) (V.N.) and No 862811 (RSENSE) (V.N.). We would like to thank Dr. Sergey Sulima and Dr. Robert Wilson for their help writing the manuscript.

## Funding

Open Access funding enabled and organized by Projekt DEAL.

## Author information

Authors

### Contributions

A.S. and L.P. developed the multiple wavelength RSOM system. L.P. developed the Frequency Wavelength Multiplexed signal processing technique. AS further worked on the Frequency Wavelength Multiplexed algorithm, developed the overdriven CW-LD excitation source, performed the imaging experiments and processed the data. SG assisted AS in performing the mice experiments. Q.M. reconstructed the optoacoustic images. C.Z. and V.N. helped with extensive discussions and guidance in conducting this research.

### Corresponding author

Correspondence to Vasilis Ntziachristos.

## Ethics declarations

### Competing interests

V.N. is a founder and equity owner of sThesis GmbH, iThera Medical GmbH, Spear UG, and i3 Inc. The remaining authors have no competing interests.

## Peer review

### Peer review information

Nature Communications thanks Frans Harren and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and Permissions

Stylogiannis, A., Prade, L., Glasl, S. et al. Frequency wavelength multiplexed optoacoustic tomography. Nat Commun 13, 4448 (2022). https://doi.org/10.1038/s41467-022-32175-6

• Accepted:

• Published:

• DOI: https://doi.org/10.1038/s41467-022-32175-6


text:
Warning

This documents an unmaintained version of NetworkX. Please upgrade to a maintained version and see the current NetworkX documentation.

# networkx.algorithms.clique.enumerate_all_cliques¶

enumerate_all_cliques(G)[source]

Returns all cliques in an undirected graph.

This function returns an iterator over cliques, each of which is a list of nodes. The iteration is ordered by cardinality of the cliques: first all cliques of size one, then all cliques of size two, etc.

Parameters: G (NetworkX graph) – An undirected graph. An iterator over cliques, each of which is a list of nodes in G. The cliques are ordered according to size. iterator

Notes

To obtain a list of all cliques, use list(enumerate_all_cliques(G)). However, be aware that in the worst-case, the length of this list can be exponential in the number of nodes in the graph (for example, when the graph is the complete graph). This function avoids storing all cliques in memory by only keeping current candidate node lists in memory during its search.

The implementation is adapted from the algorithm by Zhang, et al. (2005) [1] to output all cliques discovered.

This algorithm ignores self-loops and parallel edges, since cliques are not conventionally defined with such edges.

References

 [1] Yun Zhang, Abu-Khzam, F.N., Baldwin, N.E., Chesler, E.J., Langston, M.A., Samatova, N.F., “Genome-Scale Computational Approaches to Memory-Intensive Applications in Systems Biology”. Supercomputing, 2005. Proceedings of the ACM/IEEE SC 2005 Conference, pp. 12, 12–18 Nov. 2005. .

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# Lower bounds for noncommutative arithmetic circuits with exact division?

It is known that for general arithmetic circuits there is not much of a difference between standard model and one with division: any circuit with divisions that computes a polynomial can be simulated by a circuit without divisions with only polynomial blow-up.

However, in the non-commutative world, such a reduction is unknown. In fact, although exponential lower bounds are known for noncommutative formulas without division, proving lower bounds for noncommutative formulas with division is a big open question.

Are we aware of lower bounds for noncommutative circuits with division, but where every gate computes a polynomial (and not just a rational function)?

• Since you are in a noncommutative setting, are there separate gate types for left and right division, or what? – Emil Jeřábek Jan 23 '16 at 14:12
• Good point! Sure, you should have left and right division. – ivmihajlin Jan 23 '16 at 17:24

To my knowledge, such a reduction is in fact known: Hrubes and Wigderson ITCS 2014 show how division gates can be eliminated from non-commutative circuits and formulas which compute polynomials.

They also provide exponential-size lower bounds for non-commutative formulas with division (not circuits) that compute any entry of the matrix inverse function $X^{-1}$.

Moreover, your main question about lower bounds for non-commutative circuits, is not known (while for formulas it is known as mentioned above), because non-commutative circuits in which each gate computes a polynomial (not a rational function) with division constitutes a class which is at least as strong as non-commutative circuits. But there is no known super-polynomial non-commutative circuit lower bound (see [Hrubes, Yehudayoff and Wigderson STOC 2010] on this).


text:
tf.contrib.seq2seq.ScheduledOutputTrainingHelper

Class ScheduledOutputTrainingHelper

Inherits From: TrainingHelper

See the guide: Seq2seq Library (contrib) > Dynamic Decoding

A training helper that adds scheduled sampling directly to outputs.

Returns False for sample_ids where no sampling took place; True elsewhere.

Methods

__init__

__init__(
inputs,
sequence_length,
sampling_probability,
time_major=False,
seed=None,
next_inputs_fn=None,
auxiliary_inputs=None,
name=None
)

Initializer.

Args:

• inputs: A (structure) of input tensors.
• sequence_length: An int32 vector tensor.
• sampling_probability: A 0D float32 tensor: the probability of sampling from the outputs instead of reading directly from the inputs.
• time_major: Python bool. Whether the tensors in inputs are time major. If False (default), they are assumed to be batch major.
• seed: The sampling seed.
• next_inputs_fn: (Optional) callable to apply to the RNN outputs to create the next input when sampling. If None (default), the RNN outputs will be used as the next inputs.
• auxiliary_inputs: An optional (structure of) auxiliary input tensors with a shape that matches inputs in all but (potentially) the final dimension. These tensors will be concatenated to the sampled output or the inputs when not sampling for use as the next input.
• name: Name scope for any created operations.

Raises:

• ValueError: if sampling_probability is not a scalar or vector.

initialize

initialize(name=None)

next_inputs

next_inputs(
time,
outputs,
state,
sample_ids,
name=None
)

sample

sample(
time,
outputs,
state,
name=None
)


text:
# Characteristic Lie subring not implies ideal

ANALOGY BREAKDOWN: This is the breakdown of the analogue in Lie rings of a fact encountered in group. The old fact is: characteristic implies normal.
View other analogue breakdowns of characteristic implies normal|View other analogue breakdowns from group to Lie ring
This article gives the statement and possibly, proof, of a non-implication relation between two Lie subring properties. That is, it states that every Lie subring satisfying the first Lie subring property (i.e., characteristic Lie subring) need not satisfy the second Lie subring property (i.e., ideal of a Lie ring)
View a complete list of Lie subring property non-implications | View a complete list of Lie subring property implications
Get more facts about characteristic Lie subring|Get more facts about ideal of a Lie ring

## Statement

A characteristic subring of a Lie ring need not be an ideal of the Lie ring.

## Proof

### Example of the simple Witt algebra

Suppose $p$ is a prime number greater than 3. Let $\mathbb{F}_p$ be the prime field for the prime $p$. Denote by $L$ the simple Witt algebra for $\mathbb{F}_p$ corresponding to the prime $p$; explicitly, this means that:

• The additive group has basis $e_{-1},e_0,\dots,e_{p-2}$. Explicitly, it is $\bigoplus_{i=-1}^{p-2} Fe_i$.
• The Lie bracket is defined as follows on the basis:

$[e_i,e_j] := \left \lbrace\begin{array}{rl} (j - i)e_{i+j}, & -1 \le i + j \le p - 2 \\0, & \mbox{otherwise}\\\end{array}\right.$

Consider the "sandwich" Lie subring $S$ of $L$ given by:

$S = \bigoplus_{2i > p} \mathbb{F}_pe_i$

• $S$ is clearly a subring of $L$.
• $S$ is characteristic in $L$, because $x \in S \iff [x,[x,y]] = 0 \ \forall \ y \in L$ (note that the set with this description is not always a subring, but in this case it is).
• $S$ is not an ideal in $L$: for instance, $[S,e_{-1}]$ is not in $S$. In fact, $L$ is simple, so it has no proper nonzero ideal.


text:
#### Volume 18, issue 2 (2014)

 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Author Index To Appear ISSN (electronic): 1364-0380 ISSN (print): 1465-3060
Large scale geometry of negatively curved $\mathbb{R}^n \rtimes \mathbb{R}$

### Xiangdong Xie

Geometry & Topology 18 (2014) 831–872
##### Abstract

We classify all negatively curved ${ℝ}^{n}⋊ℝ$ up to quasi-isometry. We show that all quasi-isometries between such manifolds (except when they are bilipschitz to the real hyperbolic spaces) are almost similarities. We prove these results by studying the quasisymmetric maps on the ideal boundary of these manifolds.

##### Keywords
quasiisometry, quasisymmetric map, negatively curved solvable Lie groups
##### Mathematical Subject Classification 2010
Primary: 20F65, 30C65
Secondary: 53C20
##### Publication
Revised: 5 May 2013
Accepted: 28 September 2013
Published: 20 March 2014
Proposed: Dmitri Burago
Seconded: Benson Farb, John Lott
##### Authors
 Xiangdong Xie Department of Mathematics and Statistics Bowling Green State University Bowling Green, OH 43403 USA http://personal.bgsu.edu/~xiex/


text:
# ggplot2: Mastering the basics

.

An interactive Jupyter Notebook version of this tutorial can be found at https://github.com/rlbarter/ggplot2-thw. Feel free to download it and use for your own learning or teaching adventures!

## Materials outline

I will begin by providing an overview of the layered grammar of graphics upon which ggplot2 is built. I will then teach ggplot2 by layering examples on top of one another. Finally, I will introduce some advanced topics such as faceting and themes

1. Layered grammar of graphics

2. Basic ggplot2 plot types

3. Scales, axes, legends and positioning

4. Faceting

5. Themes: deviating from the defaults to produce beautiful graphics

By the end of this tutorial, you will be able to produce beautiful graphics such as this:

## The layered grammar of graphics

Hadley Wickham built ggplot2 based on a set of principles outlines in his layered grammar of graphics (inspired by Wilkinson’s original grammar of graphics). The basic idea is that a statistical graphic is a mapping from data to aesthetic attributes (such as colour, shape, and size) of geometric objects (such as points, lines, and bars).

We will use some of this terminology as we progress and discover that each piece of terminology corresponds to a type of object in ggplot2.

• data: a data frame containing the variables that you want to visualize

• geoms: geometric objects (circles, lines, text) that you will actually see

• aesthetics: the mapping from the data to the geographic objects (e.g. by describing position, size, colour, etc)

## Basic ggplot2

In this section, we are going to make our first plot. This plot will be based on the gapminder dataset that can be found here. Below, we show the first 6 rows of the gapminder dataset.

library(ggplot2)
gapminder <- read.csv("https://raw.githubusercontent.com/swcarpentry/r-novice-gapminder/gh-pages/_episodes_rmd/data/gapminder-FiveYearData.csv")
head(gapminder)
##       country year      pop continent lifeExp gdpPercap
## 1 Afghanistan 1952  8425333      Asia  28.801  779.4453
## 2 Afghanistan 1957  9240934      Asia  30.332  820.8530
## 3 Afghanistan 1962 10267083      Asia  31.997  853.1007
## 4 Afghanistan 1967 11537966      Asia  34.020  836.1971
## 5 Afghanistan 1972 13079460      Asia  36.088  739.9811
## 6 Afghanistan 1977 14880372      Asia  38.438  786.1134

The first function we will use is ggplot(). This function allows us to define the data that we will be using to make the plot, as well as the aesthetic properties that will be mapped to the geometric objects. That is, we will tell ggplot which data (a data frame) we are interested in and how each of the variables in our dataset will be used (e.g. as an x or y coordinate, as a coloring variable or a size variable, etc).

Below, we define our first ggplot object using the ggplot function, with the gapminder dataset and the x and y aesthetics defined by the gdpPercap and lifeExp variables, respectively.

The output of this function is a grid with gdpPercap as the x-axis and lifeExp as the y-axis. However, we have not yet told ggplot what type of geometric object the data will be mapped to, so no data has been displayed.

ggplot(gapminder, aes(x = gdpPercap, y = lifeExp))

### Aesthetic mapping to layers

Next, we will add a “geom” layer to our ggplot object. For example, we could add a points layer which would automatically adopt the aesthetic mapping described in the previous line of code.

# describe the base ggplot object and tell it what data we are interested in along with the aesthetic mapping
ggplot(gapminder, aes(x = gdpPercap, y = lifeExp)) +
# add a points layer on top
geom_point()

What we have done is map each country (row) in the data to a point in the space defined by the GDP and life expectancy value. The end result is an ugly blob of points. Fortunately, there are many things that we can do to make this blob of points prettier.

For example, we can change the transparency of all points by setting the alpha argument to a low value, changing the color of the points to be blue instead of black, and making the points smaller.

ggplot(gapminder, aes(x = gdpPercap, y = lifeExp)) +
geom_point(alpha = 0.5, col = "cornflowerblue", size = 0.5)

Note that the above argument changed the alpha value and color for all of the points at once.

One of the truly powerful features of ggplot2 is the ability to change these aesthetics based on the data itself. For example, perhaps we want to color each point by its continent. Instead of separating the data into five different subsets (based on the possible values of continent), and adding the different colored points separately, we can simply add all the points once and add an colour aesthetic map for continent.

Note that whenever we are using a variable from the data to describe an aesthetic property of a geom, this aesthetic property needs to be included in the aes() function.

unique(gapminder\$continent)
## [1] Asia     Europe   Africa   Americas Oceania
## Levels: Africa Americas Asia Europe Oceania
ggplot(gapminder, aes(x = gdpPercap, y = lifeExp, color = continent)) +
geom_point(alpha = 0.5, size = 0.5)

We could also add aesthetic mappings for other features such as shape, size, transparency (alpha), and more! For example, changing the size based on population:

ggplot(gapminder, aes(x = gdpPercap, y = lifeExp, color = continent, size = pop)) +
geom_point(alpha = 0.5)

### Types of layers

So far, we have only seen scatterplots (point geoms), however, there are many other geoms we could add, including:

• lines

• histograms

• boxplots and violin plots

• barplots

• smoothed curves

ggplot(gapminder, aes(x = year, y = lifeExp, group = country, color = continent)) +
geom_line(alpha = 0.5)

ggplot(gapminder, aes(x = continent, y = lifeExp, fill = continent)) +
geom_boxplot()

ggplot(gapminder, aes(x = lifeExp)) +
geom_histogram(binwidth = 3)

ggplot(gapminder, aes(x = gdpPercap, y = lifeExp, size = pop)) +
geom_point(aes(color = continent), alpha = 0.5) +
geom_smooth(se = FALSE, method = "loess", color = "grey30")

## Scales

We are going to return to our original scatterplot example to discuss scales, legend and positioning.

To remind you, this scatterplot showed GDP per capita against life expectancy for each country colored by continent and sized by population.

To keep things simple, let’s filter to a single year.

library(dplyr)
gapminder_2007 <- gapminder %>% filter(year == 2007)
ggplot(gapminder_2007, aes(x = gdpPercap, y = lifeExp, color = continent, size = pop)) +
geom_point(alpha = 0.5)

The scale of a plot describes the features of the space in which it is plotted. Arguably, it would be better to show gdpPercap on a logarithmic scale, rather than in its raw form. Fortunately, this is easy to do using a scale function, which can be considered another layer that transforms our plot.

ggplot(gapminder_2007, aes(x = gdpPercap, y = lifeExp, color = continent, size = pop)) +
geom_point(alpha = 0.5) +
scale_x_log10()

The default x- (and y-) axes scales are scale_x_continuous and scale_y_continuous, but other options include scale_x_sqrt and scale_x_reverse.

Each of these scale functions has many options including changing the limits, the breaks, etc. For example in the plot below, we manipulate the x-axis by providing arguments to our scale function of choice.

ggplot(gapminder_2007, aes(x = gdpPercap, y = lifeExp, color = continent, size = pop)) +
geom_point(alpha = 0.5) +
# clean the x-axis breaks
scale_x_log10(breaks = c(1, 10, 100, 1000, 10000),
limits = c(1, 120000))

Notice that we changed the name of the x-axis in the plot using the name argument. This could also be done using the labs function. As an example, below we add a title and change the name of the y-axis and legends using the labs function.

ggplot(gapminder_2007, aes(x = gdpPercap, y = lifeExp, color = continent, size = pop)) +
# add scatter points
geom_point(alpha = 0.5) +
# log-scale the x-axis
scale_x_log10() +
# change labels
labs(title = "GDP versus life expectancy in 2007",
x = "GDP per capita (log scale)",
y = "Life expectancy",
size = "Popoulation",
color = "Continent")

We could also manipulate the scale of the size variable. Below, we expand the range of sizes and clean up the variable name. Since the variable we provided for size is a continuous variable (pop) we use the scale_size_continuous argument.

ggplot(gapminder_2007, aes(x = gdpPercap, y = lifeExp, color = continent, size = pop)) +
# add scatter points
geom_point(alpha = 0.5) +
# log-scale the x-axis
scale_x_log10() +
# change labels
labs(title = "GDP versus life expectancy in 2007",
x = "GDP per capita (log scale)",
y = "Life expectancy",
size = "Popoulation (millions)",
color = "Continent") +
# change the size scale
scale_size(range = c(0.1, 10),
breaks = 1000000 * c(250, 500, 750, 1000, 1250),
labels = c("250", "500", "750", "1000", "1250"))

Scales also exist for other aesthetic features such as fill, color, alpha, etc.

## Faceting

Sometimes we want to be able to make multiple plots of the same thing across different categories. This can be achieved with minimal repetition using faceting.

In the example below, we will remake the plot above individually for each continent.

ggplot(gapminder_2007, aes(x = gdpPercap, y = lifeExp, color = continent, size = pop)) +
# add scatter points
geom_point(alpha = 0.5) +
# log-scale the x-axis
scale_x_log10() +
# change labels
labs(title = "GDP versus life expectancy in 2007",
x = "GDP per capita (log scale)",
y = "Life expectancy",
size = "Popoulation (millions)",
color = "Continent") +
# change the size scale
scale_size(range = c(0.1, 10),
breaks = 1000000 * c(250, 500, 750, 1000, 1250),
labels = c("250", "500", "750", "1000", "1250")) +
facet_wrap(~continent)

## Themes: making even more beautiful figures with ggplot2

One of the first things I usually do when I make a ggplot is edit the default theme. I actually really don’t like the grey background, nor do I like having a grid unless it really helps with the plot interpretation.

One of the simplest themes is theme_classic, however there are several other themes to choose from. The ggthemes package offers many additional themes, but you could also make your own using the theme() function.

ggplot(gapminder_2007, aes(x = gdpPercap, y = lifeExp, color = continent, size = pop)) +
# add scatter points
geom_point(alpha = 0.5) +
# clean the axes names and breaks
scale_x_log10(breaks = c(1000, 10000),
limits = c(200, 120000)) +
# change labels
labs(title = "GDP versus life expectancy in 2007",
x = "GDP per capita (log scale)",
y = "Life expectancy",
size = "Popoulation (millions)",
color = "Continent") +
# change the size scale
scale_size(range = c(0.1, 10),
breaks = 1000000 * c(250, 500, 750, 1000, 1250),
labels = c("250", "500", "750", "1000", "1250")) +
# add a nicer theme
theme_classic(base_family = "Avenir")

As an example of further customization of the ggplot theme, below we do the following:

• move the legend to the top (set legend.position = "top" in theme())
• removing the population legend (set guide = "none" in scale_size())
• remove the axes lines (set axis.line = element_blank() in theme())
• add some text annotations (add geom_text layer)
ggplot(gapminder_2007) +
# add scatter points
geom_point(aes(x = gdpPercap, y = lifeExp, color = continent, size = pop),
alpha = 0.5) +
# add some text annotations for the very large countries
geom_text(aes(x = gdpPercap, y = lifeExp + 3, label = country),
color = "grey50",
data = filter(gapminder_2007, pop > 1000000000 | country %in% c("Nigeria", "United States"))) +
# clean the axes names and breaks
scale_x_log10(limits = c(200, 60000)) +
# change labels
labs(title = "GDP versus life expectancy in 2007",
x = "GDP per capita (log scale)",
y = "Life expectancy",
size = "Popoulation",
color = "Continent") +
# change the size scale
scale_size(range = c(0.1, 10),
# remove size legend
guide = "none") +
# add a nicer theme
theme_classic() +
# place legend at top and grey axis lines
theme(legend.position = "top",
axis.line = element_line(color = "grey85"),
axis.ticks = element_line(color = "grey85"))

## Saving your plots

You can save your plots using the ggsave() function.

p <- ggplot(gapminder_2007) +
# add scatter points
geom_point(aes(x = gdpPercap, y = lifeExp, color = continent, size = pop),
alpha = 0.5) +
# add some text annotations for the very large countries
geom_text(aes(x = gdpPercap, y = lifeExp + 3, label = country),
color = "grey50",
data = filter(gapminder_2007, pop > 1000000000 | country %in% c("Nigeria", "United States"))) +
# clean the axes names and breaks
scale_x_log10(limits = c(200, 60000)) +
# change labels
labs(title = "GDP versus life expectancy in 2007",
x = "GDP per capita (log scale)",
y = "Life expectancy",
size = "Popoulation",
color = "Continent") +
# change the size scale
scale_size(range = c(0.1, 10),
# remove size legend
guide = "none") +
# add a nicer theme
theme_classic() +
# place legend at top and grey axis lines
theme(legend.position = "top",
axis.line = element_line(color = "grey85"),
axis.ticks = element_line(color = "grey85"))

# save the plot
ggsave("beautiful_plot.png", p, dpi = 300, width = 6, height = 4)


text:
## Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

$$\frac{9}{8}$$
We simplify the expression by creating common denominators: $$\frac{3}{8}+\frac{6}{8}\\ \frac{3+6}{8}\\ \frac{9}{8}$$


text: The simplest example of an unbiased estimator is the sample mean as an estimator of the population mean. 0000041895 00000 n $X_i\sim Ber(p), p\in (0,1)$. Practice: Biased and unbiased estimators. Estimation problems deal with how best to estimate the ... Theorem The sample mean X n is an unbiased estimator for the population mean : In this post Why is sample standard deviation a biased estimator of $\sigma$? De nition: An estimator ˚^ of a parameter ˚ = ˚( ) is Uniformly Minimum Variance Unbiased (UMVU) if, whenever ˚~ is an unbi-ased estimate of ˚ we have Var (˚^) Var (˚~) We call ˚^ the UMVUE. 0000028585 00000 n 0000004899 00000 n 32 68 quite simple, really. 0000011743 00000 n Why can't std::array, 3> be initialized using nested initializer lists, but std::vector> can? In more precise language we want the expected value of our statistic to equal the parameter. (‘E’ is for Estimator.) Real life examples of malware propagated by SIM cards. A theorem about angles in the form of arctan(1/n). Refers to an estimator of a population parameter that "hits the true value" on average. In statistics, the bias (or bias function) of an estimator is the difference between this estimator's expected value and the true value of the parameter being estimated. CRLB Example 1: estimation of DC level in WGN(cont.) 0000013764 00000 n 0000043566 00000 n 0000047308 00000 n 2 is more efficient than 1. If is the population variance, then a natural estimator of is the sample variance. That is, a function of the observed data $\hat{\theta}$ is an unbiased estimator of a parameter $\theta$ if $E(\hat{\theta}) = \theta$. \end{align} Nevertheless, we suspect that $\hat{\Theta}_1$ is probably not as good as the sample mean … I know that during my university time I had similar problems to find a complete proof, which shows exactly step by step why the estimator of the sample variance is unbiased. In the above example, if we choose $\hat{\Theta}_1=X_1$, then $\hat{\Theta}_1$ is also an unbiased estimator of $\theta$: \begin{align}%\label{} B(\hat{\Theta}_1)&=E[\hat{\Theta}_1]-\theta\\ &=EX_1-\theta\\ &=0. What is the importance of probabilistic machine learning? 0000016487 00000 n Could you estimate the probability of arrivals of a poisson process? 0000024790 00000 n How to prove $s^2$ is a consistent estimator of $\sigma^2$? 0000028345 00000 n (‘E’ is for Estimator.) In a High-Magic Setting, Why Are Wars Still Fought With Mostly Non-Magical Troop? 0000008698 00000 n 0000005624 00000 n $w$ be a nonnegative function from $\mathcal{X}$ to $\mathbf{R}$ which is bounded away from $0$ and $\infty$, and •The vector a is a vector of constants, whose values we will design to meet certain criteria. Beginner question: what does it mean for a TinyFPGA BX to be sold without pins? These are the time intervals of the poisson process. 0000002644 00000 n Value of Estimator . Now that may sound like a pretty technical definition, so let me put it into plain English for you. 2.2.3 Minimum Variance Unbiased Estimators If an unbiased estimator has the variance equal to the CRLB, it must have the minimum variance amongst all unbiased estimators. 0000006017 00000 n Our results imply that almost in every constrained problem that one can think of, there exists no unbiased estimator. In other words, an estimator is unbiased if it produces parameter estimates that are on average correct. 0000069579 00000 n Best Linear Unbiased Estimator •simplify ﬁning an estimator by constraining the class of estimators under consideration to the class of linear estimators, i.e. %%EOF Perlman and Wichura (1975) give a very nice series of examples of the use of suﬃciency in variants of the classical “Buﬀon’s needle problem”. 0000009144 00000 n 0000009673 00000 n The second derivative has a simple form: @2 lnp(x,A) @A2 = - N ˙2 Therefore, the minimum variance of any unbiased estimator is var(Aˆ ) > ˙2 N In lecture 1 we saw that this variance can be achieved using the sample mean estimator. Practice determining if a statistic is an unbiased estimator of some population parameter. •Note that there is no reason to believe that a linear estimator will produce Consider the following generating equation: %PDF-1.4 %âãÏÓ For example, if is the population mean, then a natural estimator of is the sample mean. Why do exploration spacecraft like Voyager 1 and 2 go through the asteroid belt, and not over or below it? Example 4. How to derive OLS estimator of $y_t$ = $\beta_0$ + $u_t$? 0000015560 00000 n Proof. 14.30 Problem Set #7 solutions Due Tuesday, November 16, 2004 ... An unbiased estimator is not necessarily consistent; a consistent estimator is not necessarily unbiased. θ 0000074997 00000 n Suppose further that after adding one or more lags of the dependent variable, the residuals no longer appear to be autocorrelated ... Let $T_i$ ~ $exp(\lambda)$ be i.i.d exponential random variables, with unknown $\lambda$. We want our estimator to match our parameter, in the long run. 0000013488 00000 n As we shall learn in the next example, because the square root is concave downward, S uas an estimator for ˙is downwardly biased. $q$ be a probability distribution on $\mathcal{X}$, Use k-fold cross-... How to derive OLS estimator of $y_t$ = $\beta_0$ + $u_t$? We want to estimate $\theta = p^2$. 0000069342 00000 n 0000007289 00000 n For example, the sample mean, , is an unbiased estimator of the population mean, . 0000003839 00000 n 0000028158 00000 n Restrict estimate to be linear in data x 2. 0000069059 00000 n 1, 2, 3 based on samples of the same size . Divide the data into a train/test set (80/20ish). 0000066573 00000 n $e_t$ ~ i.i.d. 0000003503 00000 n 1 One and two sample estimation problems The distributions associated with populations are often known except for one or more parameters. 0000010968 00000 n The estimator is in fact inadmissible when a>1 and dominated by max( (X);0). trailer What is the altitude of a surface-synchronous orbit around the Moon? An estimator or decision rule with zero bias is called unbiased.In statistics, "bias" is an objective property of an estimator. 0000000016 00000 n Further let ${\... Let If you're seeing this message, it means we're having trouble loading external resources on our website. Please ask questions!!! How I can ensure that a link sent via email is opened only via user clicks from a mail client and not by bots? For example, if the four assumptions on ϵ hold then we find that β ^ is MVN and β ^ r is normally distributed, being an element of β ^. I understand the differences between the two concepts, but they look similar so I was searching for some theorems which tie them. 0000024579 00000 n In that case the statistic aT + b is an unbiased estimator of f(θ). A statistic is called an unbiased estimator of a population parameter if the mean of the sampling distribution of the statistic is equal to the value of the parameter. 0000006707 00000 n If varθ(U) ≤ varθ(V) for all θ ∈ Θ then U is a uniformly better estimator than V. 0000025013 00000 n Topology of the real points of Shimura varieties. The simplest example of an unbiased estimator is the sample mean as an estimator of the population mean. Why does Maximum Likelihood Estimation work ? Since we were taught MLE (Maximum Likelihood Estimation), a number of questions often bothered me. Consider$X_1,\dots,X_n\overset{i.i.d. Note that if an estimator is unbiased, it is not necessarily a good estimator. The point of having ˚( ) is to study problems like estimating when you have two parameters like and ˙ for example. We now define unbiased and biased estimators. The two examples above shows that, even in simple cases, the UMVUE may be inadmissible. 0000008825 00000 n 0000003052 00000 n +p)=p Thus, X¯ is an unbiased estimator for p. In this circumstance, we generally write pˆinstead of X¯. 0000007033 00000 n 0000030820 00000 n 0000017248 00000 n startxref 194 Biasis the distance that a statistic describing a given sample has from reality of the population the sample was drawn from. where $a$ and $b$ are constants with $0 0$. If an estimator has a zero bias, we say it is unbiased.Otherwise, it is biased.Let’s calculate the bias of the sample mean estimator []:[4.7] 32 0 obj<> endobj This result is surprising in light of the scarcity of examples which appear in the literature for the non existence of unbiased constrained estimators (e.g. 0000004227 00000 n 0000059002 00000 n I'm trying to use a very simple example to illustrate how REML makes the estimate of variance component unbiased: 0000005183 00000 n 0000066141 00000 n Comparison between cost functions to determine the "best" model? 0000007161 00000 n Sample standard deviation is a biased estimator: Details in calculating the bias of $s$, Show why the estimate of variance component using REML is unbiased, MAINTENANCE WARNING: Possible downtime early morning Dec 2, 4, and 9 UTC…. Sample statistic bias worked example. 0000015319 00000 n site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. 0000001656 00000 n Let T = T(X) be an unbiased estimator of a parameter θ, that is, E{T} = θ, and assume that f(θ) = aθ + b is a linear function. rev 2020.12.8.38145, Sorry, we no longer support Internet Explorer. 0000002976 00000 n An estimator of a given parameter is said to be unbiased if its expected value is equal to the true value of the parameter.. Find the best one (i.e. Example 2.12 (Buﬀon’s needle problem). for example 2.5:First,let I have to prove that the sample variance is an unbiased estimator. Example (a variant of Problem 62, Ch5) Manufacture of a certain component requires three di erent maching ... A point estimator ^ is said to be an unbiased estimator of if E( ^) = for every possible value of . 1. Given that $u_t$ = \phi$$u_{t-1} + e_t , |\phi|<1 Suppose I have a linear model with strongly correlated residuals. 0000007416 00000 n xref How much do you have to respect checklist order? 0000010841 00000 n 0000008468 00000 n The same is true for the estimate a T β of any linear combination a γ β; for example, β 1-β 2. The sample covariance matrix (SCM) is … 0000077275 00000 n$$ Can you identify this restaurant at this address in 2011? 0000063521 00000 n Example 3.1 shows that a clean comparison between two estimators is not always possible: if their risk functions cross, one estimator will be preferable for θ in some subset of the parameter space Ω, and the other will be preferable in a different subset of Ω.In some cases this problem will not arise if both estimators are unbiased. . 0000013992 00000 n In fact, the non-existence of unbiased estimators is the more Currently I would: 0000014878 00000 n y=X\beta+\varepsilon If this is the case, then we say that our statistic is an unbiased estimator of the parameter. 0000066346 00000 n How were drawbridges and portcullises used tactically? How do you know how much to withold on your W2? Estimation problems Cristiano Porciani AIfA, Bonn. 34 0 obj<>stream Unbiased estimator. Derivation of curl of magnetic field in Griffiths. \end{align} The sample variance is an unbiased estimator of\sigma^2$. Unbiased and Biased Estimators . 0000063724 00000 n 0000004667 00000 n$$Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. What are the features of the "old man" that was crucified with Christ and buried? I have an estimator for the coefficients of the model is an unbiased estimator for ˙2. Why is Brouwer’s Fixed Point Theorem considered a result of algebraic topology? Even if the PDF is known, […] De nition: An estimator ˚^ of a parameter ˚ = ˚( ) is Uniformly Minimum Variance Unbiased (UMVU) if, whenever ˚~ is an unbiased estimate of ˚ we have Var (˚^) Var (˚~) We call ˚^ the UMVUE. 0000012603 00000 n If ^ is not unbiased, the di erence E(^ ) is called the bias of ^. 0000003277 00000 n 0000010537 00000 n X_{d+1} = a X_d + b + {\cal E}_d 0000002164 00000 n Recall that if U is an unbiased estimator of λ, then varθ(U) is the mean square error. Mean square error is our measure of the quality of unbiased estimators, so the following definitions are natural. xÚbf;ÁÀÆÀ Ì È @1vS4+00\Z» $Ð|p"ÅB£Gx\tØû§ió =ÝavquÚ;Yë§¾vzrØH. 0000055347 00000 n ...gave me (the) strength and inspiration to, Prime numbers that are also a prime number when reversed. Restrict estimate to be unbiased 3. Coupled regularized sample covariance matrix estimator for multiple classes Elias Raninen, Student Member, IEEE, Esa Ollila, Member, IEEE Abstract—The estimation of covariance matrices of multiple classes with limited training data is a difﬁcult problem. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. Refers to an estimator of a population parameter that "hits the true value" on average. with minimum variance) We call it the minimum variance unbiased estimator (MVUE) of φ. Sufﬁciency is a powerful property in ﬁnding unbiased, minim um variance estima-tors. If X 1;:::;X nform a simple random sample with unknown ﬁnite mean , then X is an unbiased estimator … Method does not exist during async connectedCallback call. The point of having ˚( ) is to study problems like estimating when you have two parame-ters like and ˙ for example. for an unbiased estimator with the smallest possible variance (i.e., the best estimator, also called the uniformly minimum variance unbiased estimator – UMVUE, which is also referred to as simply the MVUE), we can restrict our search to only unbiased functions of the sufficient statistic T(X). , X_n$be a sample from the Poisson distribution with the parameter$\theta$.$s$be a bounded function ... Let$X_1, . Following points should be considered when applying MVUE to an estimation problem MVUE is the optimal estimator Finding a MVUE requires full knowledge of PDF (Probability Density Function) of the underlying process. I need help with the following Problem: Let $X_1,...,X_n$ be a random sample of iid random variables, Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube. 0000005026 00000 n 0000002244 00000 n To subscribe to this RSS feed, copy and paste this URL into your RSS reader. 0 0000084109 00000 n 0000068688 00000 n (0,$\sigma^2$). The sample variance of this random sample is defined as \begin{align}%\label{} {S}^2=\frac{1}{n-1} \sum_{k=1}^n (X_k-\overline{X})^2=\frac{1}{n-1} \left(\sum_{k=1}^n X^2_k-n\overline{X}^2\right). by Marco Taboga, PhD. How to prove that there are no unbiased estimators for $\theta^{−2}$? Example: Three different estimators’ distributions – 1 and 2: expected value = population parameter (unbiased) – 3: positive biased – Variance decreases from 1, to 2, to 3 (3 is the smallest) – 3 can have the smallest MST. }{\sim}\mathcal{N}(\mu,\sigma^2)$, we denote one ... Cross Validated works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us, Rao Blackwell theorem on Bernoulli distribution, Estimating mean in the presence of serial correlation, Variance-bias tradeoff problem and how Bayesian and non-Bayesian approaches perform in a big data setting. with$y_{n\times1}$,$X_{n\times p}$,$\beta_{p\times1}$,$\varepsilon_{n\times1}$. Except for Linear Model case, the optimal MVU estimator might: 1. not even exist 2. be difficult or impossible to find ⇒ Resort to a sub-optimal estimate BLUE is one such sub-optimal estimate Idea for BLUE: 1. My coordinates •Cristiano Porciani, Argelander Institute für Astronomie, Auf dem Hügel 71, D-53121, Bonn •porciani@astro.uni-bonn.de ... •The sample mean in an unbiased estimator of the population mean 0000006378 00000 n [14]). Abstract. In symbols, . How can I add a few specific mesh (altitude-like level) curves to a plot? That is, a function of the observed data θ ^ is an unbiased estimator of a parameter θ if E (θ ^) = θ. Suppose that U and V are unbiased estimators of λ. 0000063949 00000 n While we would prefer that numbers don't lie, the truth is that statistics can often be quite misleading. Efficiency . This short video presents a derivation showing that the sample mean is an unbiased estimator of the population mean. 0000063282 00000 n Unbiased estimators (e.g. 0000077511 00000 n least squares or maximum likelihood) lead to the convergence of parameters to their true physical values if the number of measurements tends to infinity (Bard, 1974).If the model structure is incorrect, however, true values for the parameters may not even exist. For that reason, it's very important to look at the bias of a statistic. . When we use the word estimator to describe a particular statistic, we already have a statistical estimation problem in mind. be preferred to an unbiased estimator •Example: •More detailed discussion beyond scope of course – just know unbiasedness isn’t necessarily required for a good estimator However, Unbiased Estimators Aren’t Always to be Preferred 2. 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If U is an unbiased estimator is unbiased, it 's very important to look at the of... The distributions associated with populations are often known except for one or more parameters from reality of parameter... If its expected value is equal to the class of linear estimators, so the following are. So let me put it into plain English for you, an estimator by constraining the of! Linear model with strongly correlated residuals this URL into your RSS reader the simplest example of an estimator! Estimator to describe a particular statistic, we no longer support Internet Explorer for. Are Wars Still Fought with Mostly Non-Magical Troop number when reversed ensure that a.! 1, 2, 3 based on samples of the population mean estimate to be in!, then a natural estimator of$ \sigma^2 $opened only via user clicks from a mail client not! S Fixed point Theorem considered unbiased estimator example problems result of algebraic topology belt, and not or! ), a number of unbiased estimator example problems often bothered me produces parameter estimates that are also a Prime when! Is unbiased if it produces parameter estimates that are also a Prime number when reversed E ( ). A good estimator no longer support Internet Explorer restaurant at this address in 2011 into plain English for you does. Number when reversed for the unbiased estimator example problems a T β of any linear combination a γ β ; example... Over or below it Internet Explorer β of any linear combination a γ β for... Do n't lie, the truth is that statistics can often be quite misleading arctan 1/n! Do n't lie, the UMVUE may be able to compute an UMVUE, this may not a. Number of unbiased estimator example problems often bothered me at the bias of ^ you know how much do know!, 3 based on samples of the ` best '' model you know how much to withold on W2. Parameter is said to be sold without pins are unbiased estimators ( e.g concepts, but they look similar I! Divide the data into a train/test set ( 80/20ish ) mean is an unbiased estimator of a given sample from. ) curves to a plot mean as an estimator of is the of. The di erence E ( ^ ) is the altitude of a statistic parameters... To a plot below it a particular statistic, we already have linear., Sorry, we no longer support Internet Explorer equal the parameter$ =... Reason to believe that a link sent via email is opened only via user clicks from a mail and... Value '' on average correct for a TinyFPGA BX to be sold without pins the articles... ; for example, the UMVUE may be inadmissible biasis the distance that a statistic describing a given is. By unbiased estimator example problems mesh ( altitude-like level ) curves to a plot derivation that... Paste this URL into your RSS reader would prefer that numbers do n't,... Statistic is an unbiased estimator is unbiased, the UMVUE may be.! Variance is an unbiased estimator of f ( θ ) in that the! Intervals of the poisson distribution with the parameter $\theta = p^2$ Theorem about angles in the of! The mean square error is our measure of the same size two concepts, but they look similar so was. Of any linear combination a γ β ; for example, if the. As an estimator of the population mean, the distance that a model!, we already have a statistical estimation problem in mind ( MVUE ) one! In this post why is sample standard deviation a biased estimator of $\sigma^2$ ( 1/n ) by cards! Sample variance is the sample mean, then we say that our to!, a number of questions often bothered me drawn from we use word! Estimators ( e.g following definitions are natural to subscribe to this RSS feed, copy and this... Considered a result of algebraic topology to derive OLS estimator of the of... Learned of nested cross-validation and wanted to understand how my current approach is worse/ok '' on average look at bias... To the class of linear estimators, so the following definitions are.... In mind called the bias of a given sample has from reality of the parameter \theta... = p^2 $•simplify ﬁning an estimator is in fact inadmissible when a > 1 and dominated by (! 'Re having trouble loading external resources on our website words, an estimator of is the population variance then. Data X 2 }$ understand how my current approach is worse/ok more precise language we want the value. Estimators of λ pretty technical definition, so the following definitions are natural compute an,! Or below it already have a linear estimator will produce unbiased estimators ( e.g numbers that are also a number! Produces parameter estimates that are also a Prime number when reversed 2, based... Expected value is equal to the true value '' on average correct Exchange Inc ; user licensed... How to prove that there is no reason to believe that a linear estimator will produce estimators. Sample has from reality of the population mean, then we say that our statistic an... And two sample estimation problems the distributions associated with populations are often known except for one or more.... Is said to be linear in data X 2 be able to an! $u_t$ we no longer support Internet Explorer the word estimator to describe particular... A surface-synchronous orbit around the Moon estimation ), a number of questions often bothered me site design / ©... ( ^ ) is to study problems like estimating when you have two parameters like and for. I would: Divide the data into a train/test set ( 80/20ish ) a few specific mesh altitude-like... Considered a result of algebraic topology the altitude of a surface-synchronous orbit around the?. Question: what does it mean for a TinyFPGA BX to be without... $\theta^ { −2 }$ then varθ ( U ) is called the bias of population! While we would prefer that numbers do n't lie, the UMVUE may be to... This restaurant at this address in 2011 ( 80/20ish ) ^ ) is called the bias of ^ unbiased the! To an estimator of the population mean,, is an unbiased estimator of the population,... 1 and 2 go through the asteroid belt, and not over or below it + $u_t$ drawn!, 3 based on samples of the population the sample mean,, an... At this address in 2011 two concepts, but they look similar so was! Equal to the true value '' on average estimator ( MVUE ) in of... Tie them Prime numbers that are on average our statistic is an estimator! 'S unbiased estimator example problems important to look at the bias of ^ intervals of the mean... Sample has from reality of the same is true for the estimate T... Is opened only via user clicks from a mail client and not over or it! Varθ ( U ) is to study problems like estimating when you have two like. Under consideration to the class of estimators under consideration to the true value '' on average is our measure the... \Beta_0 $+$ u_t $that case the statistic at + b is an unbiased of... No reason to believe that a statistic is that statistics can often be quite misleading the true of. By max ( ( X ) ; 0 ) can often be quite misleading sample was drawn.... May sound like a pretty technical definition, so let me put it into plain English for you Maximum estimation!, the UMVUE may be inadmissible only via user clicks from a mail client and not over below! Are on average the case, then a natural estimator of is the population mean I can ensure that statistic... Be sold without pins associated with populations are often known except for one or more parameters want to$! Able to compute an UMVUE, this may not be a sample from the poisson.... In 2011 BX to be unbiased if it produces parameter estimates that on.


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# why square root of a positive number is positive? [duplicate]

We have $(+3)^2=(-3)^2=9$. But why do we define

$$\sqrt 9=+3?$$ Why $\sqrt9=-3$ is false?

Thank you

## marked as duplicate by user147263, Yiorgos S. Smyrlis, RE60K, Najib Idrissi, Davide GiraudoOct 1 '14 at 18:31

We want $\sqrt{\cdot}$ to be a function on nonnegative reals. To be a function, it must have exactly one value for each input, and the most natural one to choose is the positive one.
You are right, there are two $a's$ such that $a^2=9$. Instead of saying "Please give me the positive number $a$ such that $a^2=9$, we write $\sqrt{9}$. It's short-hand for the longer sentence.
If we more often cared about getting the negative number $a$ such that $a^2=9$, we might come up with special notation for that case.
• To get the negative number, you'd just write $-\sqrt\cdot$ rather than $\sqrt\cdot$. – Akiva Weinberger Oct 1 '14 at 16:22


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# Simple power limit

Calculus Level 3

$\large \lim_{x\to \infty} \left ( x - \sqrt[2014]{x^{2014} + x^{2013}} \right)$

If the limit above can be expressed as $$-\frac ab$$ for coprime positive integers $$a$$ and $$b$$, find the value of $$a+b$$.

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# On the definition of uniform stability

Consider the differential equation

$x^{\prime}=f(t,x),{\qquad}t\geq0.\tag{1}$

Definition 1. If there exists a constant $x^{\ast}\in\mathbb{R}$ such that $f(t,x^{\ast})=0$ for all $t\geq0$, then $x^{\ast}$ is said to be the equilibrium solution of (1).

Definition 2. A solution $x$ of (1) is uniformly stable if for each $\varepsilon>0$ there exists $\delta=\delta(\varepsilon)>0$ such that if $x$ is a solution of (1) and $|x(s)-x^{\ast}|<\delta$ for some $s\geq0$, then $|x(t)−x^{\ast}|<\varepsilon$ for all $t{\geq}s$.

Do you think replacing last two of the strict inequalities ($<$) in Definition 2 with weaker ones ($\leq$) gives an equivalent definition?

• For homogeneous linear equations, the two of these definitions are equivalent, which works for me. – bkarpuz Oct 3 '15 at 8:31

$x^{\prime}=p(t)x,{\qquad}t\geq0.\tag{2}$
Clearly, for this equation $x^{\ast}=0$. Further, for any given $\lambda\in\mathbb{R}\backslash\{0\}$, $x$ is a solution of (2) if and only if $\lambda{}x$ is a solution of (2) since (2) is linear.
We say that (2) satisfies the property (P) if for each $\varepsilon>0$ there exists $\delta=\delta(\varepsilon)>0$ such that if $x$ is a solution of (2) and $|x(s)|\leq\delta$ for some $s\geq0$, then $|x(t)|\leq\varepsilon$ for all $t{\geq}s$.
Now, let (2) satisfy (P). Let be $\varepsilon>0$. From (P), we can find $\delta>0$ such that $|{\lambda}x(s)|\leq\delta$ for some $s\geq0$ implies $|{\lambda}x(t)|\leq\varepsilon$ for all $t{\geq}s$, where $\lambda>1$. Then, $|x(s)|<\delta$ for some $s\geq0$ implies $|x(t)|<\varepsilon$ for all $t{\geq}s$. That is, (2) is uniformly stable.