gaussian_process_guider.cpp

/*
    PHD2 Guiding
 
    @file
    @date      2014-2017
    @copyright Max Planck Society
 
    @brief     Provides a Gaussian process based guiding algorithm
 
    SPDX-FileCopyrightText: Edgar D. Klenske <edgar.klenske@tuebingen.mpg.de>
    SPDX-FileCopyrightText: Stephan Wenninger <stephan.wenninger@tuebingen.mpg.de>
    SPDX-FileCopyrightText: Raffi Enficiaud <raffi.enficiaud@tuebingen.mpg.de>
 
    SPDX-License-Identifier: BSD-3-Clause
*/
 
#include "gaussian_process_guider.h"
 
#include <cmath>
#include <ctime>
#include <iostream>
#include <iomanip>
#include <fstream>
 
// HY: Added this include and a few qCDebug EKos logging calls below.
#include "ekos_guide_debug.h"
 
#define DLOG if (false) qCDebug
 
#define SAVE_FFT_DATA_ 0
#define PRINT_TIMINGS_ 0
 
#define CIRCULAR_BUFFER_SIZE 8192 // for the raw data storage
#define REGULAR_BUFFER_SIZE 2048 // for the regularized data storage
#define FFT_SIZE 4096 // for zero-padding the FFT, >= REGULAR_BUFFER_SIZE!
#define GRID_INTERVAL 5.0
#define MAX_DITHER_STEPS 10 // for our fallback dithering
 
#define DEFAULT_LEARNING_RATE 0.01 // for a smooth parameter adaptation
 
#define HYSTERESIS 0.1 // for the hybrid mode
 
GaussianProcessGuider::GaussianProcessGuider(guide_parameters parameters) :
    start_time_(std::chrono::system_clock::now()),
    last_time_(std::chrono::system_clock::now()),
    control_signal_(0),
    prediction_(0),
    last_prediction_end_(0),
    dither_steps_(0),
    dithering_active_(false),
    dither_offset_(0.0),
    circular_buffer_data_(CIRCULAR_BUFFER_SIZE),
    covariance_function_(),
    output_covariance_function_(),
    gp_(covariance_function_),
    learning_rate_(DEFAULT_LEARNING_RATE),
    parameters(parameters)
{
    circular_buffer_data_.push_front(data_point()); // add first point
    circular_buffer_data_[0].control = 0; // set first control to zero
    gp_.enableExplicitTrend(); // enable the explicit basis function for the linear drift
    gp_.enableOutputProjection(output_covariance_function_); // for prediction
 
    std::vector<double> hyperparameters(NumParameters);
    hyperparameters[SE0KLengthScale] = parameters.SE0KLengthScale_;
    hyperparameters[SE0KSignalVariance] = parameters.SE0KSignalVariance_;
    hyperparameters[PKLengthScale] = parameters.PKLengthScale_;
    hyperparameters[PKSignalVariance] = parameters.PKSignalVariance_;
    hyperparameters[SE1KLengthScale] = parameters.SE1KLengthScale_;
    hyperparameters[SE1KSignalVariance] = parameters.SE1KSignalVariance_;
    hyperparameters[PKPeriodLength] = parameters.PKPeriodLength_;
    SetGPHyperparameters(hyperparameters);
 
    qCDebug(KSTARS_EKOS_GUIDE) <<
                               QString("GPG Parameters: control_gain %1 min_move %2 pred_gain %3 min_for_inf %4 min_for_period %5 pts %6 cpd %7 -- se0L %8 se0V %9 PL %10 PV %11 Se1L %12 se1V %13 ppd %14")
                               .arg(parameters.control_gain_, 6, 'f', 3)
                               .arg(parameters.min_move_, 6, 'f', 3)
                               .arg(parameters.prediction_gain_, 6, 'f', 3)
                               .arg(parameters.min_periods_for_inference_, 6, 'f', 3)
                               .arg(parameters.min_periods_for_period_estimation_, 6, 'f', 3)
                               .arg(parameters.points_for_approximation_)
                               .arg(parameters.compute_period_)
                               .arg(parameters.SE0KLengthScale_, 6, 'f', 3)
                               .arg(parameters.SE0KSignalVariance_, 6, 'f', 3)
                               .arg(parameters.PKLengthScale_, 6, 'f', 3)
                               .arg(parameters.PKSignalVariance_, 6, 'f', 3)
                               .arg(parameters.SE1KLengthScale_, 6, 'f', 3)
                               .arg(parameters.SE1KSignalVariance_, 6, 'f', 3)
                               .arg(parameters.PKPeriodLength_, 6, 'f', 3);
}
 
GaussianProcessGuider::~GaussianProcessGuider()
{
}
 
void GaussianProcessGuider::SetTimestamp()
{
    auto current_time = std::chrono::system_clock::now();
    double delta_measurement_time = std::chrono::duration<double>(current_time - last_time_).count();
    last_time_ = current_time;
    get_last_point().timestamp = std::chrono::duration<double>(current_time - start_time_).count()
                                 - (delta_measurement_time / 2.0) // use the midpoint as time stamp
                                 + dither_offset_; // correct for the gear time offset from dithering
}
 
// adds a new measurement to the circular buffer that holds the data.
void GaussianProcessGuider::HandleGuiding(double input, double SNR)
{
    SetTimestamp();
    get_last_point().measurement = input;
    get_last_point().variance = CalculateVariance(SNR);
 
    // we don't want to predict for the part we have measured!
    // therefore, don't use the past when a measurement is available.
    last_prediction_end_ = get_last_point().timestamp;
}
 
void GaussianProcessGuider::HandleDarkGuiding()
{
    SetTimestamp();
    get_last_point().measurement = 0; // we didn't actually measure
    get_last_point().variance = 1e4; // add really high noise
}
 
void GaussianProcessGuider::HandleControls(double control_input)
{
    get_last_point().control = control_input;
}
 
double GaussianProcessGuider::CalculateVariance(double SNR)
{
    SNR = std::max(SNR, 3.4); // limit the minimal SNR
 
    // this was determined by simulated experiments
    double standard_deviation = 2.1752 * 1 / (SNR - 3.3) + 0.5;
 
    return standard_deviation * standard_deviation;
}
 
void GaussianProcessGuider::UpdateGP(double prediction_point /*= std::numeric_limits<double>::quiet_NaN()*/)
{
#if PRINT_TIMINGS_
    clock_t begin = std::clock(); // this is for timing the method in a simple way
#endif
 
    size_t N = get_number_of_measurements();
 
    // initialize the different vectors needed for the GP
    Eigen::VectorXd timestamps(N - 1);
    Eigen::VectorXd measurements(N - 1);
    Eigen::VectorXd variances(N - 1);
    Eigen::VectorXd sum_controls(N - 1);
 
    double sum_control = 0;
 
    // transfer the data from the circular buffer to the Eigen::Vectors
    for (size_t i = 0; i < N - 1; i++)
    {
        sum_control += circular_buffer_data_[i].control; // sum over the control signals
        timestamps(i) = circular_buffer_data_[i].timestamp;
        measurements(i) = circular_buffer_data_[i].measurement;
        variances(i) = circular_buffer_data_[i].variance;
        sum_controls(i) = sum_control; // store current accumulated control signal
    }
 
    Eigen::VectorXd gear_error(N - 1);
    Eigen::VectorXd linear_fit(N - 1);
 
    // calculate the accumulated gear error
    gear_error = sum_controls + measurements; // for each time step, add the residual error
 
#if PRINT_TIMINGS_
    clock_t end = std::clock();
    double time_init = double(end - begin) / CLOCKS_PER_SEC;
    begin = std::clock();
#endif
 
    // regularize the measurements
    Eigen::MatrixXd result = regularize_dataset(timestamps, gear_error, variances);
 
    // the three vectors are returned in a matrix, we need to extract them
    timestamps = result.row(0);
    gear_error = result.row(1);
    variances = result.row(2);
 
#if PRINT_TIMINGS_
    end = std::clock();
    double time_regularize = double(end - begin) / CLOCKS_PER_SEC;
    begin = std::clock();
#endif
 
    // linear least squares regression for offset and drift to de-trend the data
    Eigen::MatrixXd feature_matrix(2, timestamps.rows());
    feature_matrix.row(0) = Eigen::MatrixXd::Ones(1, timestamps.rows()); // timestamps.pow(0)
    feature_matrix.row(1) = timestamps.array(); // timestamps.pow(1)
 
    // this is the inference for linear regression
    Eigen::VectorXd weights = (feature_matrix * feature_matrix.transpose()
                               + 1e-3 * Eigen::Matrix<double, 2, 2>::Identity()).ldlt().solve(feature_matrix * gear_error);
 
    // calculate the linear regression for all datapoints
    linear_fit = weights.transpose() * feature_matrix;
 
    // subtract polynomial fit from the data points
    Eigen::VectorXd gear_error_detrend = gear_error - linear_fit;
 
#if PRINT_TIMINGS_
    end = std::clock();
    double time_detrend = double(end - begin) / CLOCKS_PER_SEC;
    begin = std::clock();
    double time_fft = 0; // need to initialize in case the FFT isn't calculated
#endif
 
    // calculate period length if we have enough points already
    double period_length = GetGPHyperparameters()[PKPeriodLength];
    if (GetBoolComputePeriod() && get_last_point().timestamp > parameters.min_periods_for_period_estimation_ * period_length)
    {
        // find periodicity parameter with FFT
        period_length = EstimatePeriodLength(timestamps, gear_error_detrend);
        UpdatePeriodLength(period_length);
 
#if PRINT_TIMINGS_
        end = std::clock();
        time_fft = double(end - begin) / CLOCKS_PER_SEC;
#endif
    }
 
#if PRINT_TIMINGS_
    begin = std::clock();
#endif
 
    // inference of the GP with the new points, maximum accuracy should be reached around current time
    gp_.inferSD(timestamps, gear_error, parameters.points_for_approximation_, variances, prediction_point);
 
#if PRINT_TIMINGS_
    end = std::clock();
    double time_gp = double(end - begin) / CLOCKS_PER_SEC;
 
    printf("timings: init: %f, regularize: %f, detrend: %f, fft: %f, gp: %f, total: %f\n",
           time_init, time_regularize, time_detrend, time_fft, time_gp,
           time_init + time_regularize + time_detrend + time_fft + time_gp);
#endif
}
 
double GaussianProcessGuider::PredictGearError(double prediction_location)
{
    // in the first step of each sequence, use the current time stamp as last prediction end
    if (last_prediction_end_ < 0.0)
    {
        last_prediction_end_ = std::chrono::duration<double>(std::chrono::system_clock::now() - start_time_).count();
    }
 
    // prediction from the last endpoint to the prediction point
    Eigen::VectorXd next_location(2);
    next_location << last_prediction_end_, prediction_location + dither_offset_;
    Eigen::VectorXd prediction = gp_.predictProjected(next_location);
 
    double p1 = prediction(1);
    double p0 = prediction(0);
 
    assert(!math_tools::isNaN(p1 - p0));
 
    last_prediction_end_ = next_location(1); // store current endpoint
 
    // we are interested in the error introduced by the gear over the next time step
    return (p1 - p0);
}
 
double GaussianProcessGuider::result(double input, double SNR, double time_step, double prediction_point /*= -1*/)
{
    /*
     * Dithering behaves differently from pausing. During dithering, the mount
     * is moved and thus we can assume that we applied the perfect control, but
     * we cannot trust the measurement. Once dithering has settled, we can trust
     * the measurement again and we can pretend nothing has happend.
     */
    double hyst_percentage = 0.0;
    double period_length;
 
    if (dithering_active_)
    {
        DLOG(KSTARS_EKOS_GUIDE) << QString("GPG::result(in=%1,snr=%2,ts=%3,ppt=%4) dithering active")
                                .arg(input, 6, 'f', 2).arg(SNR, 5, 'f', 1).arg(time_step, 6, 'f', 2).arg(prediction_point, 5, 'f', 1);
        if (--dither_steps_ <= 0)
        {
            dithering_active_ = false;
        }
        deduceResult(time_step); // just pretend we would do dark guiding...
 
        GPDebug->Log("PPEC rslt(dithering): input = %.2f, final = %.2f",
                     input, parameters.control_gain_ * input);
 
        DLOG(KSTARS_EKOS_GUIDE)
                << QString("PPEC rslt(dithering): input = %1, final = %2")
                .arg(input, 5, 'f', 2)
                .arg(parameters.control_gain_ * input, 5, 'f', 2);
 
        return parameters.control_gain_ * input; // ...but apply proportional control
    }
 
    DLOG(KSTARS_EKOS_GUIDE) << QString("GPG::result(in=%1,snr=%2,ts=%3,ppt=%4)")
                            .arg(input, 6, 'f', 2).arg(SNR, 5, 'f', 1).arg(time_step, 6, 'f', 2).arg(prediction_point, 5, 'f', 1);
    // the starting time is set at the first call of result after startup or reset
    if (get_number_of_measurements() == 1)
    {
        start_time_ = std::chrono::system_clock::now();
        last_time_ = start_time_; // this is OK, since last_time_ only provides a minor correction
    }
 
    // collect data point content, except for the control signal
    HandleGuiding(input, SNR);
 
    // calculate hysteresis result, too, for hybrid control
    double last_control = 0.0;
    if (get_number_of_measurements() > 1)
    {
        last_control = get_second_last_point().control;
    }
    double hysteresis_control = (1.0 - HYSTERESIS) * input + HYSTERESIS * last_control;
    hysteresis_control *= parameters.control_gain_;
 
    control_signal_ = parameters.control_gain_ * input; // start with proportional control
    if (std::abs(input) < parameters.min_move_)
    {
        control_signal_ = 0.0; // don't make small moves
        hysteresis_control = 0.0;
    }
    assert(std::abs(control_signal_) == 0.0 || std::abs(input) >= parameters.min_move_);
 
    // calculate GP prediction
    if (get_number_of_measurements() > 10)
    {
        if (prediction_point < 0.0)
        {
            prediction_point = std::chrono::duration<double>(std::chrono::system_clock::now() - start_time_).count();
        }
        // the point of highest precision shoud be between now and the next step
        UpdateGP(prediction_point + 0.5 * time_step);
 
        // the prediction should end after one time step
        prediction_ = PredictGearError(prediction_point + time_step);
        control_signal_ += parameters.prediction_gain_ * prediction_; // add the prediction
 
        // smoothly blend over between hysteresis and GP
        period_length = GetGPHyperparameters()[PKPeriodLength];
        if (get_last_point().timestamp < parameters.min_periods_for_inference_ * period_length)
        {
            double percentage = get_last_point().timestamp / (parameters.min_periods_for_inference_ * period_length);
            percentage = std::min(percentage, 1.0); // limit to 100 percent GP
            hyst_percentage = 1.0 - percentage;
            control_signal_ = percentage * control_signal_ + (1.0 - percentage) * hysteresis_control;
        }
    }
    else
    {
        period_length = GetGPHyperparameters()[PKPeriodLength]; // for logging
    }
 
    // assert for the developers...
    assert(!math_tools::isNaN(control_signal_));
 
    // ...safeguard for the users
    if (math_tools::isNaN(control_signal_))
    {
        control_signal_ = hysteresis_control;
    }
 
    add_one_point(); // add new point here, since the control is for the next point in time
    HandleControls(control_signal_); // already store control signal
 
    GPDebug->Log("PPEC rslt: input = %.2f, final = %.2f, react = %.2f, pred = %.2f, hyst = %.2f, hyst_pct = %.2f, period_length = %.2f",
                 input, control_signal_, parameters.control_gain_ * input, parameters.prediction_gain_ * prediction_, hysteresis_control,
                 hyst_percentage, period_length);
 
    DLOG(KSTARS_EKOS_GUIDE) <<
                            QString("PPEC rslt: input = %1, final = %2, react = %3, pred = %4, hyst = %5, hyst_pct = %6, period_length = %7")
                            .arg(input, 5, 'f', 2)
                            .arg(control_signal_, 5, 'f', 2)
                            .arg(parameters.control_gain_ * input, 5, 'f', 2)
                            .arg(parameters.prediction_gain_ * prediction_, 5, 'f', 2)
                            .arg(hysteresis_control, 5, 'f', 2)
                            .arg(hyst_percentage, 5, 'f', 2)
                            .arg(period_length, 5, 'f', 2);
    return control_signal_;
}
 
double GaussianProcessGuider::deduceResult(double time_step, double prediction_point /*= -1.0*/)
{
    HandleDarkGuiding();
 
    control_signal_ = 0; // no measurement!
    // check if we are allowed to use the GP
    if (get_number_of_measurements() > 10
            && get_last_point().timestamp > parameters.min_periods_for_inference_ * GetGPHyperparameters()[PKPeriodLength])
    {
        if (prediction_point < 0.0)
        {
            prediction_point = std::chrono::duration<double>(std::chrono::system_clock::now() - start_time_).count();
        }
        // the point of highest precision should be between now and the next step
        UpdateGP(prediction_point + 0.5 * time_step);
 
        // the prediction should end after one time step
        prediction_ = PredictGearError(prediction_point + time_step);
        control_signal_ += prediction_; // control based on prediction
    }
 
    add_one_point(); // add new point here, since the control is for the next point in time
    HandleControls(control_signal_); // already store control signal
 
    // assert for the developers...
    assert(!math_tools::isNaN(control_signal_));
 
    // ...safeguard for the users
    if (math_tools::isNaN(control_signal_))
    {
        control_signal_ = 0.0;
    }
 
    return control_signal_;
}
 
void GaussianProcessGuider::reset()
{
    qCDebug(KSTARS_EKOS_GUIDE) << QString("GPG::reset()");
    circular_buffer_data_.clear();
    gp_.clearData();
 
    // We need to add a first data point because the measurements are always relative to the control.
    // For the first measurement, we therefore need to add a point with zero control.
    circular_buffer_data_.push_front(data_point()); // add first point
    circular_buffer_data_[0].control = 0; // set first control to zero
 
    last_prediction_end_ = -1.0; // the negative value signals we didn't predict yet
    start_time_ = std::chrono::system_clock::now();
    last_time_ = std::chrono::system_clock::now();
 
    dither_offset_ = 0.0;
    dither_steps_ = 0;
    dithering_active_ = false;
}
 
void GaussianProcessGuider::GuidingDithered(double amt, double rate)
{
    DLOG(KSTARS_EKOS_GUIDE) << QString("GPG::GuidingDithered(amt=%1,rate=%2)")
                            .arg(amt, 6, 'f', 2).arg(rate, 5, 'f', 1);
 
    // we store the amount of dither in seconds of gear time
    dither_offset_ += amt / rate; // this is the amount of time offset
 
    dithering_active_ = true;
    dither_steps_ = MAX_DITHER_STEPS;
}
 
void GaussianProcessGuider::GuidingDitherSettleDone(bool success)
{
    DLOG(KSTARS_EKOS_GUIDE) << QString("GPG::GuidingDitherSettleDone(%1)")
                            .arg(success ? "true" : "false");
    if (success)
    {
        dither_steps_ = 1; // the last dither step should always be executed by
        // result(), since it corrects for the time difference
    }
}
 
void GaussianProcessGuider::DirectMoveApplied(double amt, double rate)
{
    (void) amt; // silence compiler warning while the code below is commented out (HY).
    (void) rate;
    // we store the amount of dither in seconds of gear time
    // todo: validate this:
    // dither_offset_ += amt / rate; // this is the amount of time offset
}
 
double GaussianProcessGuider::GetControlGain(void) const
{
    return parameters.control_gain_;
}
 
bool GaussianProcessGuider::SetControlGain(double control_gain)
{
    parameters.control_gain_ = control_gain;
    return false;
}
 
bool GaussianProcessGuider::GetBoolComputePeriod() const
{
    return parameters.compute_period_;
}
 
bool GaussianProcessGuider::SetBoolComputePeriod(bool active)
{
    parameters.compute_period_ = active;
    return false;
}
 
std::vector<double> GaussianProcessGuider::GetGPHyperparameters() const
{
    // since the GP class works in log space, we have to exp() the parameters first.
    Eigen::VectorXd hyperparameters_full = gp_.getHyperParameters().array().exp();
    // remove first parameter, which is unused here
    Eigen::VectorXd hyperparameters = hyperparameters_full.tail(NumParameters);
 
    // converts the length-scale of the periodic covariance from standard notation to natural units
    hyperparameters(PKLengthScale) = std::asin(hyperparameters(PKLengthScale) / 4.0) * hyperparameters(PKPeriodLength) / M_PI;
 
    // we need to map the Eigen::vector into a std::vector.
    return std::vector<double>(hyperparameters.data(), // the first element is at the array address
                               hyperparameters.data() + NumParameters);
}
 
bool GaussianProcessGuider::SetGPHyperparameters(std::vector<double> const &hyperparameters)
{
    Eigen::VectorXd hyperparameters_eig = Eigen::VectorXd::Map(&hyperparameters[0], hyperparameters.size());
 
    // prevent length scales from becoming too small (makes GP unstable)
    hyperparameters_eig(SE0KLengthScale) = std::max(hyperparameters_eig(SE0KLengthScale), 1.0);
    hyperparameters_eig(PKLengthScale) = std::max(hyperparameters_eig(PKLengthScale), 1.0);
    hyperparameters_eig(SE1KLengthScale) = std::max(hyperparameters_eig(SE1KLengthScale), 1.0);
 
    // converts the length-scale of the periodic covariance from natural units to standard notation
    hyperparameters_eig(PKLengthScale) = 4 * std::sin(hyperparameters_eig(PKLengthScale)
                                         * M_PI / hyperparameters_eig(PKPeriodLength));
 
    // safeguard all parameters from being too small (log conversion)
    hyperparameters_eig = hyperparameters_eig.array().max(1e-10);
 
    // need to convert to GP parameters
    Eigen::VectorXd hyperparameters_full(NumParameters + 1); // the GP has one more parameter!
    hyperparameters_full << 1.0, hyperparameters_eig;
 
    // the GP works in log space, therefore we need to convert
    gp_.setHyperParameters(hyperparameters_full.array().log());
    return false;
}
 
double GaussianProcessGuider::GetMinMove() const
{
    return parameters.min_move_;
}
 
bool GaussianProcessGuider::SetMinMove(double min_move)
{
    parameters.min_move_ = min_move;
    return false;
}
 
int GaussianProcessGuider::GetNumPointsForApproximation() const
{
    return parameters.points_for_approximation_;
}
 
bool GaussianProcessGuider::SetNumPointsForApproximation(int num_points)
{
    parameters.points_for_approximation_ = num_points;
    return false;
}
 
double GaussianProcessGuider::GetPeriodLengthsInference() const
{
    return parameters.min_periods_for_inference_;
}
 
bool GaussianProcessGuider::SetPeriodLengthsInference(double num_periods)
{
    parameters.min_periods_for_inference_ = num_periods;
    return false;
}
 
double GaussianProcessGuider::GetPeriodLengthsPeriodEstimation() const
{
    return parameters.min_periods_for_period_estimation_;
}
 
bool GaussianProcessGuider::SetPeriodLengthsPeriodEstimation(double num_periods)
{
    parameters.min_periods_for_period_estimation_ = num_periods;
    return false;
}
 
double GaussianProcessGuider::GetPredictionGain() const
{
    return parameters.prediction_gain_;
}
 
bool GaussianProcessGuider::SetPredictionGain(double prediction_gain)
{
    parameters.prediction_gain_ = prediction_gain;
    return false;
}
 
void GaussianProcessGuider::inject_data_point(double timestamp, double input, double SNR, double control)
{
    // collect data point content, except for the control signal
    HandleGuiding(input, SNR);
    last_prediction_end_ = timestamp;
    get_last_point().timestamp = timestamp; // overrides the usual HandleTimestamps();
 
    start_time_ = std::chrono::system_clock::now() - std::chrono::seconds((int) timestamp);
 
    add_one_point(); // add new point here, since the control is for the next point in time
    HandleControls(control); // already store control signal
}
 
double GaussianProcessGuider::EstimatePeriodLength(const Eigen::VectorXd &time, const Eigen::VectorXd &data)
{
    // compute Hamming window to reduce spectral leakage
    Eigen::VectorXd windowed_data = data.array() * math_tools::hamming_window(data.rows()).array();
 
    // compute the spectrum
    std::pair<Eigen::VectorXd, Eigen::VectorXd> result = math_tools::compute_spectrum(windowed_data, FFT_SIZE);
 
    Eigen::ArrayXd amplitudes = result.first;
    Eigen::ArrayXd frequencies = result.second;
 
    double dt = (time(time.rows() - 1) - time(0)) / (time.rows() - 1); // (t_end - t_begin) / num_t
 
    frequencies /= dt; // correct for the average time step width
 
    Eigen::ArrayXd periods = 1 / frequencies.array();
    amplitudes = (periods > 1500.0).select(0, amplitudes); // set amplitudes to zero for too large periods
 
    assert(amplitudes.size() == frequencies.size());
 
    Eigen::VectorXd::Index maxIndex;
    amplitudes.maxCoeff(&maxIndex);
 
    double max_frequency = frequencies(maxIndex);
 
    // quadratic interpolation to find maximum
    // check if we can interpolate
    if (maxIndex < frequencies.size() - 1 && maxIndex > 0)
    {
        double spread = std::abs(frequencies(maxIndex - 1) - frequencies(maxIndex + 1));
 
        Eigen::VectorXd interp_loc(3);
        interp_loc << frequencies(maxIndex - 1), frequencies(maxIndex), frequencies(maxIndex + 1);
        interp_loc = interp_loc.array() - max_frequency; // centering for numerical stability
        interp_loc = interp_loc.array() / spread; // normalize for numerical stability
 
        Eigen::VectorXd interp_dat(3);
        interp_dat << amplitudes(maxIndex - 1), amplitudes(maxIndex), amplitudes(maxIndex + 1);
        interp_dat = interp_dat.array() / amplitudes(maxIndex); // normalize for numerical stability
 
        // we need to handle the case where all amplitudes are equal
        // the linear regression would be unstable in this case
        if (interp_dat.maxCoeff() - interp_dat.minCoeff() < 1e-10)
        {
            return 1 / max_frequency; // don't do the linear regression
        }
 
 
        // building feature matrix
        Eigen::MatrixXd phi(3, 3);
        phi.row(0) = interp_loc.array().pow(2);
        phi.row(1) = interp_loc.array().pow(1);
        phi.row(2) = interp_loc.array().pow(0);
 
        // standard equation for linear regression
        Eigen::VectorXd w = (phi * phi.transpose()).ldlt().solve(phi * interp_dat);
 
        // recovering the maximum from the weights relative to the frequency of the maximum
        max_frequency = max_frequency - w(1) / (2 * w(0)) * spread; // note the de-normalization
    }
 
#if SAVE_FFT_DATA_
    {
        std::ofstream outfile;
        outfile.open("spectrum_data.csv", std::ios_base::out);
        if (outfile)
        {
            outfile << "period, amplitude\n";
            for (int i = 0; i < amplitudes.size(); ++i)
            {
                outfile << std::setw(8) << periods[i] << "," << std::setw(8) << amplitudes[i] << "\n";
            }
        }
        else
        {
            std::cout << "unable to write to file" << std::endl;
        }
        outfile.close();
    }
#endif
 
    double period_length = 1 / max_frequency; // we return the period length!
    return period_length;
}
 
void GaussianProcessGuider::UpdatePeriodLength(double period_length)
{
    std::vector<double> hypers = GetGPHyperparameters();
 
    // assert for the developers...
    assert(!math_tools::isNaN(period_length));
 
    // ...and save the day for the users
    if (math_tools::isNaN(period_length))
    {
        period_length = hypers[PKPeriodLength]; // just use the old value instead
    }
 
    // we just apply a simple learning rate to slow down parameter jumps
    hypers[PKPeriodLength] = (1 - learning_rate_) * hypers[PKPeriodLength] + learning_rate_ * period_length;
 
    SetGPHyperparameters(hypers); // the setter function is needed to convert parameters
}
 
Eigen::MatrixXd GaussianProcessGuider::regularize_dataset(const Eigen::VectorXd &timestamps,
        const Eigen::VectorXd &gear_error, const Eigen::VectorXd &variances)
{
    size_t N = get_number_of_measurements();
    double grid_interval = GRID_INTERVAL;
    double last_cell_end = -grid_interval;
    double last_timestamp = -grid_interval;
    double last_gear_error = 0.0;
    double last_variance = 0.0;
    double gear_error_sum = 0.0;
    double variance_sum = 0.0;
    int grid_size = static_cast<int>(std::ceil(timestamps(timestamps.size() - 1) / grid_interval)) + 1;
    assert(grid_size > 0);
    Eigen::VectorXd reg_timestamps(grid_size);
    Eigen::VectorXd reg_gear_error(grid_size);
    Eigen::VectorXd reg_variances(grid_size);
    int j = 0;
    for (size_t i = 0; i < N - 1; ++i)
    {
        if (timestamps(i) < last_cell_end + grid_interval)
        {
            gear_error_sum += (timestamps(i) - last_timestamp) * 0.5 * (last_gear_error + gear_error(i));
            variance_sum += (timestamps(i) - last_timestamp) * 0.5 * (last_variance + variances(i));
            last_timestamp = timestamps(i);
        }
        else
        {
            while (timestamps(i) >= last_cell_end + grid_interval)
            {
                double inter_timestamp = last_cell_end + grid_interval;
 
                double proportion = (inter_timestamp - last_timestamp) / (timestamps(i) - last_timestamp);
                double inter_gear_error = proportion * gear_error(i) + (1 - proportion) * last_gear_error;
                double inter_variance = proportion * variances(i) + (1 - proportion) * last_variance;
 
                gear_error_sum += (inter_timestamp - last_timestamp) * 0.5 * (last_gear_error + inter_gear_error);
                variance_sum += (inter_timestamp - last_timestamp) * 0.5 * (last_variance + inter_variance);
 
                reg_timestamps(j) = last_cell_end + 0.5 * grid_interval;
                reg_gear_error(j) = gear_error_sum / grid_interval;
                reg_variances(j) = variance_sum / grid_interval;
 
                last_timestamp = inter_timestamp;
                last_gear_error = inter_gear_error;
                last_variance = inter_variance;
                last_cell_end = inter_timestamp;
 
                gear_error_sum = 0.0;
                variance_sum = 0.0;
                ++j;
            }
        }
    }
    if (j > REGULAR_BUFFER_SIZE)
    {
        j = REGULAR_BUFFER_SIZE;
    }
 
    // We need to output 3 vectors. For simplicity, we join them into a matrix.
    Eigen::MatrixXd result(3, j);
    result.row(0) = reg_timestamps.head(j);
    result.row(1) = reg_gear_error.head(j);
    result.row(2) = reg_variances.head(j);
 
    return result;
}
 
void GaussianProcessGuider::save_gp_data() const
{
    // write the GP output to a file for easy analyzation
    size_t N = get_number_of_measurements();
 
    if (N < 2)
    {
        return; // cannot save data before first measurement
    }
 
    // initialize the different vectors needed for the GP
    Eigen::VectorXd timestamps(N - 1);
    Eigen::VectorXd measurements(N - 1);
    Eigen::VectorXd variances(N - 1);
    Eigen::VectorXd controls(N - 1);
    Eigen::VectorXd sum_controls(N - 1);
    Eigen::VectorXd gear_error(N - 1);
    Eigen::VectorXd linear_fit(N - 1);
 
    // transfer the data from the circular buffer to the Eigen::Vectors
    for (size_t i = 0; i < N - 1; i++)
    {
        timestamps(i) = circular_buffer_data_[i].timestamp;
        measurements(i) = circular_buffer_data_[i].measurement;
        variances(i) = circular_buffer_data_[i].variance;
        controls(i) = circular_buffer_data_[i].control;
        sum_controls(i) = circular_buffer_data_[i].control;
        if(i > 0)
        {
            sum_controls(i) += sum_controls(i - 1); // sum over the control signals
        }
    }
    gear_error = sum_controls + measurements; // for each time step, add the residual error
 
    int M = 512; // number of prediction points
    Eigen::VectorXd locations = Eigen::VectorXd::LinSpaced(M, 0, get_second_last_point().timestamp + 1500);
 
    Eigen::VectorXd vars(locations.size());
    Eigen::VectorXd means = gp_.predictProjected(locations, &vars);
    Eigen::VectorXd stds = vars.array().sqrt();
 
    {
        std::ofstream outfile;
        outfile.open("measurement_data.csv", std::ios_base::out);
        if(outfile)
        {
            outfile << "location, output\n";
            for( int i = 0; i < timestamps.size(); ++i)
            {
                outfile << std::setw(8) << timestamps[i] << "," << std::setw(8) << gear_error[i] << "\n";
            }
        }
        else
        {
            std::cout << "unable to write to file" << std::endl;
        }
        outfile.close();
 
        outfile.open("gp_data.csv", std::ios_base::out);
        if(outfile)
        {
            outfile << "location, mean, std\n";
            for( int i = 0; i < locations.size(); ++i)
            {
                outfile << std::setw(8) << locations[i] << "," << std::setw(8) << means[i] << "," << std::setw(8) << stds[i] << "\n";
            }
        }
        else
        {
            std::cout << "unable to write to file" << std::endl;
        }
        outfile.close();
    }
}
 
void GaussianProcessGuider::SetLearningRate(double learning_rate)
{
    learning_rate_ = learning_rate;
    return;
}
 
// Debug Log interface ======
 
class NullDebugLog : public GPDebug
{
        void Log(const char *fmt, ...) override
        {
            (void) fmt;
        }
};
 
class GPDebug *GPDebug = new NullDebugLog();
 
namespace
{
// just so the leak checker does not complain
struct GPDebugCleanup
{
    ~GPDebugCleanup()
    {
        GPDebug::SetGPDebug(nullptr);
    }
} s_cleanup;
}
 
void GPDebug::SetGPDebug(GPDebug *logger)
{
    delete ::GPDebug;
    ::GPDebug = logger;
}
 
GPDebug::~GPDebug()
{
}