Tuesday, November 27, 2018

Betting market at new lows for the Coalition

House Coalition Odds ($) Labor Odds ($) Coalition Win Probability (%)
2018-11-27 BetEasy 4.50 1.18 20.77
2018-11-27 Ladbrokes 3.50 1.25 26.32
2018-11-27 Sportsbet 4.25 1.16 21.44
2018-11-27 William Hill 4.25 1.20 22.02

Saturday, November 24, 2018

Leave one out

Ipsos has intrigued me for a while. My intuition was that it was noisier than the other polling firms. And so I was wondering, is this actually the case? And if it is the case, how should I reflect that in the polling aggregation model?

Normally, when thinking about the accuracy of polling houses we would look at past performance. However, most Australian pollsters are relatively new. Even the long-standing names have been subject to mergers and acquisitions over recent years. There is just not enough history to come to a clear view about polling accuracy.

Rather than look to the historical performance of the polling houses, I wondered whether - for each pollster - we could compare them with the aggregation of all the polls excluding that pollster (a leave one out or LOO analysis). We can then look at the Root Mean Square Error (RMSE) for each pollster compared with their peers.

Rather than remove each house from the model one-by-one, I changed the model so that polls from the "left-one-out" pollster reflected the population outcome with a standard deviation of 10 percentage points (roughly equivalent to a poll with a sample size of 30) . The other polls were treated as if they had a standard deviation of 1.6 percentage points (or a sample size of 1000).

This largely removed the influence of the LOO pollster in establishing the aggregate. However, it meant that we could calculate a median house bias, which we applied before assessing the degree of error for each polling house (after all, I was interested in noisiness of a pollster, not its unadjusted house bias). My medium term intention is to pare back the influence of noisy pollsters on the model.

Let's look at the analysis for each pollster. being treated as uninfluential. On these charts, LOO stands for the "left one out".

Yes the differences are small. But they are also meaningful. For example, the inclusion of Ipsos drives a dip in the analysis immediately before the Turnbull/Morrison change over. No other pollster has such a dip.

Turning to the RMSE analysis: this supports the hypothesis that Ipsos is noisier than the other polling houses. For each poll I took the difference between bias adjusted poll result and the aggregate estimate on the day of the poll (in percentage points). N is the number of polls for each pollster. The mean is the mean difference. I was expecting the mean difference to be close to zero, with the differences normally distributed around zero.  The standard deviation is the classical standard deviation for all of the differences. RMSE is the square root of the mean squared differences.

N mean difference std dev RMSE
Essential 70.0 -0.095937 0.885413 0.884211
Ipsos 12.0 -0.423814 1.497985 1.556784
Newspoll2 21.0 -0.147582 1.129458 1.139059
Roy Morgan 5.0 0.290956 1.047105 1.086777
ReachTEL 13.0 -0.001811 0.932127 0.932129
YouGov 9.0 0.284016 1.210048 1.242933
Newspoll 25.0 0.021968 0.897288 0.897557

Because I am a Bayesian, I can use the difference data and a prior normal model to reconstruct the distribution of differences for each pollster. This yields the following parameter distributions for the difference between bias adjusted poll results and the aggregate population estimate. Again, Ipsos appears to be the noisiest of the pollsters.

For completeness, the Stan code I used in estimating the mean (mu) and standard deviation (sigma) for the difference in bias adjusted poll results when compared with the aggregate estimate from the other pollsters follows.

// STAN: Bayesian Fit a Normal Distribution
data {
    int<lower=1> N;
    vector[N] y;
parameters {
    real<lower=0> sigma;
    real mu;
model {
    sigma ~ cauchy(0, 5);
    mu ~ cauchy(0, 5);
    y ~ normal(mu, sigma);

In terms of the model, for the more noisy pollsters, I plan to increase the noise term in the model (using the vector poll_qual_adj). In effect these pollsters will be treated as having a smaller sample size. The updated model follows.

// STAN: Two-Party Preferred (TPP) Vote Intention Model 
//     - Updated to allow for a discontinuity event (Turnbull -> Morrison)
//     - Updated to exclude some houses from the sum to zero constraint
//     - Updated to reduce the influence of the noisier polling houses

data {
    // data size
    int<lower=1> n_polls;
    int<lower=1> n_days;
    int<lower=1> n_houses;
    // assumed standard deviation for all polls
    real<lower=0> pseudoSampleSigma;
    // poll data
    vector<lower=0,upper=1>[n_polls] y; // TPP vote share
    int<lower=1> house[n_polls];
    int<lower=1> day[n_polls];
    vector<lower=0> [n_polls] poll_qual_adj; // poll quality adjustment
    // period of discontinuity event
    int<lower=1,upper=n_days> discontinuity;
    int<lower=1,upper=n_days> stability;
    // exclude final n houses from the house
    // effects sum to zero constraint.
    int<lower=0> n_exclude;

transformed data {
    // fixed day-to-day standard deviation
    real sigma = 0.0015;
    real sigma_volatile = 0.0045;
    int<lower=1> n_include = (n_houses - n_exclude);

parameters {
    vector[n_days] hidden_vote_share; 
    vector[n_houses] pHouseEffects;

transformed parameters {
    vector[n_houses] houseEffect;
    houseEffect[1:n_houses] = pHouseEffects[1:n_houses] - 

model {
    // -- temporal model [this is the hidden state-space model]
    hidden_vote_share[1] ~ normal(0.5, 0.15); // PRIOR
    hidden_vote_share[discontinuity] ~ normal(0.5, 0.15); // PRIOR
    hidden_vote_share[2:(discontinuity-1)] ~ 
        normal(hidden_vote_share[1:(discontinuity-2)], sigma);
    hidden_vote_share[(discontinuity+1):stability] ~ 
        normal(hidden_vote_share[discontinuity:(stability-1)], sigma_volatile);
    hidden_vote_share[(stability+1):n_days] ~ 
        normal(hidden_vote_share[stability:(n_days-1)], sigma);
    // -- house effects model
    pHouseEffects ~ normal(0, 0.08); // PRIOR 

    // -- observed data / measurement model
    y ~ normal(houseEffect[house] + hidden_vote_share[day], 
        pseudoSampleSigma + poll_qual_adj);

The updated charts follow.

Tuesday, November 20, 2018

Morrison's best polls ever

Scott Morrison has just had his two best polls ever: Ipsos and Essential have both given him 48 per cent of the two party preferred vote. Although the Bayesian model has moved in the direction of the Coalition, it needs sustained polling at this level before it would be fully convinced that his vote share is actually 48 per cent.

Turning to the moving averages, which I use to cross-check the Bayesian model, we can see they are all over the place with the latest polls. It is the same message: You need more than two polls at a 48 per cent to be convinced this is the new normal (particularly in the context of the most recent Newspoll at 45 per cent).

The primary vote Bayesian models are next.

And the possible TPP from the primary models ...

Saturday, November 17, 2018

Updated primary vote share model

For the 2019 election, I have explored a number of models for aggregating the primary vote shares, including models based on Dirichlet processes and centered logits. These older models were complicated and slow. They included constraints to ensure vote shares always summed to 100 per cent for every sample from the posterior distribution.

The model I currently run is very simple. It is based on four independent Gaussian processes for each primary vote series: Coalition, Labor, Greens and Others. In this regard it is very similar to the two-party-preferred (TPP) model.

The model has few internal constraints and it runs reasonably fast (in about 3 and half minutes). However the parameters are sampled independently, and only sum to 100 per cent in terms of the mean/median for each parameter. For the improved speed (and the absence of pesky diagnostics) this was a worthwhile compromise.

The Stan program includes a generated quantities code block in which we convert primary vote intentions to an estimated TPP vote share, based on preference flows at previous elections.

While the TPP model operates in the range 0 to 1 (with vote share values typically between 0.45 and 0.55), the new model centers all of the primary vote observations around 100. This ensures that for each party the analysis of their vote shares are well away from the edge of valid values for all model parameters. If we did not do this, the Greens vote share (often around 0.1) would be too close to the zero parameter boundary. Stan can get grumpy if it is being asked to estimate a parameter close to a boundary. 

The key outputs from the new model follow. We will start with the primary vote share aggregation and an estimate of house effects for each party.

We can compare the primary vote shares for the major and minor parties.

And we can estimate the TPP vote shares for the Coalition and Labor, based on the preference flows from previous elections

The code for the latest model is as follows.
// STAN: Primary Vote Intention Model
// Essentially a set of independent Gaussian processes from day-to-day
// for each party's primary vote, centered around a mean of 100

data {
    // data size
    int<lower=1> n_polls;
    int<lower=1> n_days;
    int<lower=1> n_houses;
    int<lower=1> n_parties;
    real<lower=0> pseudoSampleSigma;
    // Centreing factors 
    real<lower=0> center;
    real centreing_factors[n_parties];
    // poll data
    real<lower=0> centered_obs_y[n_parties, n_polls]; // poll data
    int<lower=1,upper=n_houses> house[n_polls]; // polling house
    int<lower=1,upper=n_days> poll_day[n_polls]; // day on which polling occurred

    //exclude final n parties from the sum-to-zero constraint for houseEffects
    int<lower=0> n_exclude;
    // period of discontinuity and subsequent increased volatility event
    int<lower=1,upper=n_days> discontinuity; // start with a discontinuity
    int<lower=1,upper=n_days> stability; // end - stability restored
    // day-to-day change
    real<lower=0> sigma;
    real<lower=0> sigma_volatile;

    // TPP preference flows
    vector<lower=0,upper=1>[n_parties] preference_flows_2010;
    vector<lower=0,upper=1>[n_parties] preference_flows_2013;
    vector<lower=0,upper=1>[n_parties] preference_flows_2016;

transformed data {
    int<lower=1> n_include = (n_houses - n_exclude);

parameters {
    matrix[n_days, n_parties] centre_track;
    matrix[n_houses, n_parties] pHouseEffects;

transformed parameters {
    matrix[n_houses, n_parties] houseEffects;
    for(p in 1:n_parties) {
        houseEffects[1:n_houses, p] = pHouseEffects[1:n_houses, p] - 
            mean(pHouseEffects[1:n_include, p]);

    for (p in 1:n_parties) {
        // -- house effects model
        pHouseEffects[, p] ~ normal(0, 8.0); // weakly informative PRIOR
        // -- temporal model - with a discontinuity followed by increased volatility
        centre_track[1, p] ~ normal(center, 15); // weakly informative PRIOR
        centre_track[2:(discontinuity-1), p] ~ 
            normal(centre_track[1:(discontinuity-2), p], sigma);
        centre_track[discontinuity, p] ~ normal(center, 15); // weakly informative PRIOR
        centre_track[(discontinuity+1):stability, p] ~ 
            normal(centre_track[discontinuity:(stability-1), p], sigma_volatile);
        centre_track[(stability+1):n_days, p] ~ 
            normal(centre_track[stability:(n_days-1), p], sigma);

        // -- observational model
        centered_obs_y[p,] ~ normal(houseEffects[house, p] + 
            centre_track[poll_day, p], pseudoSampleSigma);

generated quantities {
    matrix[n_days, n_parties]  hidden_vote_share;
    vector [n_days] tpp2010;
    vector [n_days] tpp2013;
    vector [n_days] tpp2016;
    for (p in 1:n_parties) {
        hidden_vote_share[,p] = centre_track[,p] - centreing_factors[p];
    // aggregated TPP estimates based on past preference flows
    for (d in 1:n_days){
        // note matrix transpose in next three lines
        tpp2010[d] = sum(hidden_vote_share'[,d] .* preference_flows_2010);
        tpp2013[d] = sum(hidden_vote_share'[,d] .* preference_flows_2013);
        tpp2016[d] = sum(hidden_vote_share'[,d] .* preference_flows_2016);
The Python program to run this Stan model follows. 
# PYTHON: analyse primary poll data

import pandas as pd
import numpy as np
import pystan
import pickle

import sys
sys.path.append( '../bin' )
from stan_cache import stan_cache

# --- check version information
print('Python version: {}'.format(sys.version))
print('pystan version: {}'.format(pystan.__version__))

# --- curate the data for the model
# key settings
intermediate_data_dir = "./Intermediate/" # analysis saved here

# preference flows
parties  =              ['L/NP', 'ALP', 'GRN', 'OTH']
preference_flows_2010 = [0.9975, 0.0, 0.2116, 0.5826]
preference_flows_2013 = [0.9975, 0.0, 0.1697, 0.5330]
preference_flows_2016 = [0.9975, 0.0, 0.1806, 0.5075]
n_parties = len(parties)

# polling data
workbook = pd.ExcelFile('./Data/poll-data.xlsx')
df = workbook.parse('Data')

# drop pre-2016 election data
df['MidDate'] = [pd.Period(d, freq='D') for d in df['MidDate']]
df = df[df['MidDate'] > pd.Period('2016-07-04', freq='D')] 

# push One Nation into Other 
df['ONP'] = df['ONP'].fillna(0)
df['OTH'] = df['OTH'] + df['ONP']

# set start date
start = df['MidDate'].min() - 1 # the first date is day 1
df['Day'] = df['MidDate'] - start # day number for each poll
n_days = df['Day'].max() # maximum days 
n_polls = len(df)

# set discontinuity date - Turnbull's last day in office
discontinuity = pd.Period('2018-08-23', freq='D') - start # UPDATE
stability = pd.Period('2018-10-01', freq='D') - start # UPDATE

# manipulate polling data ... 
y = df[parties]
center = 100
centreing_factors = center - y.mean()
y = y + centreing_factors

# add polling house data to the mix
# make sure the "sum to zero" exclusions are 
# last in the list
houses = df['Firm'].unique().tolist()
exclusions = ['YouGov', 'Ipsos']
# Note: we are excluding YouGov and Ipsos 
# from the sum to zero constraint because 
# they have unusual poll results compared 
# with other pollsters
for e in exclusions:
    assert(e in houses)
houses = houses + exclusions
map = dict(zip(houses, range(1, len(houses)+1)))
df['House'] = df['Firm'].map(map)
n_houses = len(df['House'].unique())
n_exclude = len(exclusions)

# sample metrics
sampleSize = 1000 # treat all polls as being of this size
pseudoSampleSigma = np.sqrt((50 * 50) / sampleSize) 

# --- compile model

# get the STAN model 
with open ("./Models/primary simultaneous model.stan", "r") as f:
    model = f.read()

# encode the STAN model in C++ 
sm = stan_cache(model_code=model)

# --- fit the model to the data
ct_init = np.full([n_days, n_parties], center*1.0)
def initfun():
    return dict(centre_track=ct_init)

chains = 5
iterations = 2000
data = {
        'n_days': n_days,
        'n_polls': n_polls,
        'n_houses': n_houses,
        'n_parties': n_parties,
        'pseudoSampleSigma': pseudoSampleSigma,
        'centreing_factors': centreing_factors,
        'centered_obs_y': y.T, 
        'poll_day': df['Day'].values.tolist(),
        'house': df['House'].values.tolist(), 
        'n_exclude': n_exclude,
        'center': center,
        'discontinuity': discontinuity,
        'stability': stability,
        # let's set the day-to-day smoothing 
        'sigma': 0.15,
        'sigma_volatile': 0.4,
        # preference flows at past elections
        'preference_flows_2010': preference_flows_2010,
        'preference_flows_2013': preference_flows_2013,
        'preference_flows_2016': preference_flows_2016
fit = sm.sampling(data=data, iter=iterations, chains=chains, 
    init=initfun, control={'max_treedepth':13})
results = fit.extract()

# --- check diagnostics
print('Stan Finished ...')
import pystan.diagnostics as psd

# --- save the analysis
with open(intermediate_data_dir + 'cat-' +
    'output-primary-zero-sum.pkl', 'wb') as f:
        centreing_factors, exclusions], f)

Saturday, November 10, 2018

Updated poll aggregates

I have not done this for a while ... so here goes.

First the two-party preferred (TPP) aggregation. While there was substantial movement associated with the leadership change (late August), and some volatility thereafter during September, things have settled down to around or just below where they were for the Coalition at its worst under the former Prime Minister.

Note: for the TPP aggregation, all pollsters are included in the constraint of setting house effects such that they sum to zero.

A quick comparison with my house-effect adjusted moving average models (which don't have discontinuities built in).

Since upgrading to Stan 2.18, my former primary vote model stopped working. So I have simplified and refactored this model. As a result of the refactoring, the requirement that the four parties total to 100 per cent on every day is not longer enforced in the model. Notwithstanding this, the total is typically within in 0.2 percentage points of 100 per cent (a mean of 99.99 per cent and a standard deviation of  0.11 percentage points). 

I have also taken YouGov and Ipsos out of the sum-to-zero constraint for house effects (because they were producing primary vote estimates that differed substantially from other pollsters). As a consequence, these excluded pollsters only inform the shape of the aggregation. Their polls do not inform the vertical positioning of the aggregate on the charts.

Of note:
  • The leadership turmoil in late August and into September harmed the Coalition's primary vote, and other parties primary vote and helped the primary vote share for Labor and the Greens.
  • After the volatility of the leadership change, that is to say since the start of October, Labor's primary vote has strengthened; while the Coalition's primary vote has been stagnant. 

The medians from the above plots can be compared. The following plots have the medians adjusted so that they sum to 100 per cent exactly on each and every day.

And finally, a quick look at One Nation (which is included in "other" above). Note: I have excluded Ipsos from this analysis as its treatment of One Nation is inconsistent over time when compared with the other pollsters.